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A regular or meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956201.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956202.png" /> of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956203.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956204.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956206.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956207.png" /> is a one-to-one mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956208.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u0956209.png" />. The inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562010.png" /> is then also univalent. Multivalent functions (cf. [[Multivalent function|Multivalent function]]), and in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562011.png" />-valent functions, are a generalization of univalent functions.
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In the study of univalent functions one of the fundamental problems is whether there exists a univalent mapping from a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562012.png" /> onto a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562013.png" />. A necessary condition for the existence of such a mapping is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562015.png" /> have equal degrees of connectivity (see, for example, [[#References|[1]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562017.png" /> are simply-connected domains whose boundaries contain more than one point, then this condition is also sufficient (see [[Riemann theorem|Riemann theorem]]), and the problem reduces to mapping a given domain onto a disc. In this connection, a special role is played in the theory of univalent functions on simply-connected domains by the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562019.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562020.png" /> that are regular and univalent on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562021.png" />, normalized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562023.png" />, and having the expansion
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A regular or meromorphic function  $  f $
 +
in a domain  $  B $
 +
of the extended complex plane  $  \overline{\mathbf C}\; $
 +
such that  $  f ( z _ {1} ) \neq f ( z _ {2} ) $
 +
whenever  $  z _ {1} \neq z _ {2} $,
 +
$  z _ {1} , z _ {2} \in B $,
 +
that is,  $  f $
 +
is a one-to-one mapping from  $  B $
 +
into  $  \overline{\mathbf C}\; $.  
 +
The inverse function  $  z = f ^ { - 1 } ( w) $
 +
is then also univalent. Multivalent functions (cf. [[Multivalent function|Multivalent function]]), and in particular  $  p $-
 +
valent functions, are a generalization of univalent functions.
  
In the case of multiply-connected domains, mappings of a given multiply-connected domain onto so-called canonical domains are studied (see [[Conformal mapping|Conformal mapping]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562026.png" /> be the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562027.png" /> that are meromorphic and univalent on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562028.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562029.png" />, and having an expansion
+
In the study of univalent functions one of the fundamental problems is whether there exists a univalent mapping from a given domain $  B $
 +
onto a given domain  $  B  ^  \prime  $.
 +
A necessary condition for the existence of such a mapping is that  $  B $
 +
and  $  B  ^  \prime  $
 +
have equal degrees of connectivity (see, for example, [[#References|[1]]]). If  $  B $
 +
and  $  B  ^  \prime  $
 +
are simply-connected domains whose boundaries contain more than one point, then this condition is also sufficient (see [[Riemann theorem|Riemann theorem]]), and the problem reduces to mapping a given domain onto a disc. In this connection, a special role is played in the theory of univalent functions on simply-connected domains by the class $  S $
 +
of functions $  f $
 +
that are regular and univalent on the disc  $  \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $,
 +
normalized by the conditions  $  f ( 0) = 0 $,
 +
$  f ^ { \prime } ( 0) = 1 $,  
 +
and having the expansion
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
f ( z)  = z + c _ {2} z  ^ {2} + \dots + c _ {n} z  ^ {n} + \dots ,\ \
 +
z \in \Delta .
 +
$$
  
in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562032.png" />, then this class is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562034.png" />.
+
In the case of multiply-connected domains, mappings of a given multiply-connected domain onto so-called canonical domains are studied (see [[Conformal mapping|Conformal mapping]]). Let  $  \Sigma ( B) $
 +
be the class of functions  $  F $
 +
that are meromorphic and univalent on a domain  $  B $
 +
containing the point  $  \infty $,
 +
and having an expansion
 +
 
 +
$$ \tag{2 }
 +
F ( z)  =  z + \alpha _ {0} + \alpha _ {1} z  ^ {-} 1 + \dots
 +
+ \alpha _ {n} z  ^ {-} n + \dots
 +
$$
 +
 
 +
in a neighbourhood of $  \infty $.  
 +
If $  B = \{ {z \in \overline{\mathbf C}\; } : {| z | > 1 } \} = \Delta  ^  \prime  $,  
 +
then this class is denoted by $  \Sigma $.
  
 
The basic problems in the theory of univalent functions are the following: 1) the study of the correspondence of boundaries under conformal mapping (see [[Boundary correspondence (under conformal mapping)|Boundary correspondence (under conformal mapping)]]; [[Limit elements|Limit elements]]; [[Attainable boundary point|Attainable boundary point]]); 2) obtaining [[Univalency conditions|univalency conditions]]; and 3) the solution of various extremal problems in function theory, in particular obtaining bounds for various functionals and for the range of values of functionals (see below) and systems of them in some class or other.
 
The basic problems in the theory of univalent functions are the following: 1) the study of the correspondence of boundaries under conformal mapping (see [[Boundary correspondence (under conformal mapping)|Boundary correspondence (under conformal mapping)]]; [[Limit elements|Limit elements]]; [[Attainable boundary point|Attainable boundary point]]); 2) obtaining [[Univalency conditions|univalency conditions]]; and 3) the solution of various extremal problems in function theory, in particular obtaining bounds for various functionals and for the range of values of functionals (see below) and systems of them in some class or other.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562035.png" /> is some class (set) of regular or meromorphic functions, and suppose that a complex functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562036.png" /> (or system of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562037.png" />) is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562038.png" />. The range of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562039.png" /> (or of the system of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562040.png" />) on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562041.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562042.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562044.png" /> (respectively, the set in points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562046.png" />-dimensional complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562047.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562048.png" />. Real-valued functionals are also considered. Any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562049.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562050.png" /> is called a majorant domain of the functional (or of the system of functionals). Knowledge of the range of values of a functional enables one to reduce the solution of a number of extremal problems to simple problems in analysis. For example, if the range of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562051.png" /> is known for the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562054.png" /> fixed), then the problem of finding upper and lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562055.png" /> reduces to finding the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562056.png" /> farthest from and closest to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562057.png" />.
+
Suppose that $  K $
 +
is some class (set) of regular or meromorphic functions, and suppose that a complex functional $  w = \phi ( f  ) $(
 +
or system of functionals $  \{ \phi _ {k} ( f  ) \} _ {k=} 1  ^ {n} $)  
 +
is given on $  K $.  
 +
The range of values of the functional $  \phi ( f  ) $(
 +
or of the system of functionals $  \{ \phi _ {k} ( f  ) \} _ {k=} 1  ^ {n} $)  
 +
on the class $  K $
 +
is the set $  D $
 +
of points $  w = \phi ( f  ) $
 +
in $  \mathbf C $(
 +
respectively, the set in points $  ( \phi _ {1} ( f  ) \dots \phi _ {n} ( f  )) $
 +
in $  n $-
 +
dimensional complex space $  \mathbf C  ^ {n} $)  
 +
such that $  f \in K $.  
 +
Real-valued functionals are also considered. Any set $  D  ^  \prime  $
 +
containing $  D $
 +
is called a majorant domain of the functional (or of the system of functionals). Knowledge of the range of values of a functional enables one to reduce the solution of a number of extremal problems to simple problems in analysis. For example, if the range of values $  D $
 +
is known for the functional $  f ( z _ {0} ) $,  
 +
$  f \in K $(
 +
$  z _ {0} $
 +
fixed), then the problem of finding upper and lower bounds for $  | f ( z _ {0} ) | $
 +
reduces to finding the points of $  D $
 +
farthest from and closest to the point $  w = 0 $.
  
The first substantial results in the theory of univalent functions were obtained using the [[Area principle|area principle]]. With the aid of the outer area theorem (1916), L. Bieberbach obtained precise upper and lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562059.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562060.png" /> (see [[Distortion theorems|Distortion theorems]]), gave the bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562061.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562062.png" /> and conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562063.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562064.png" /> (see [[Bieberbach conjecture|Bieberbach conjecture]]; [[Coefficient problem|Coefficient problem]]). He also found the exact value of the Koebe constant. Bounds were found for the modulus of a function and its derivative, as well as other bounds for the classes of convex, star-like, typically-real, etc., functions (cf. [[Convex function (of a complex variable)|Convex function (of a complex variable)]]; [[Star-like function|Star-like function]]; [[Typically-real function|Typically-real function]]). The [[Convexity radius|convexity radius]] and the radius of  "star-likeness"  were found for a number of classes (see [[Limit of star-likeness|Limit of star-likeness]]).
+
The first substantial results in the theory of univalent functions were obtained using the [[Area principle|area principle]]. With the aid of the outer area theorem (1916), L. Bieberbach obtained precise upper and lower bounds for $  | f ( z) | $
 +
and $  | f ^ { \prime } ( z) | $
 +
for $  f \in S $(
 +
see [[Distortion theorems|Distortion theorems]]), gave the bound $  | c _ {2} | \leq  2 $
 +
for $  f \in S $
 +
and conjectured that $  | c _ {n} | \leq  n $
 +
for $  f \in S $(
 +
see [[Bieberbach conjecture|Bieberbach conjecture]]; [[Coefficient problem|Coefficient problem]]). He also found the exact value of the Koebe constant. Bounds were found for the modulus of a function and its derivative, as well as other bounds for the classes of convex, star-like, typically-real, etc., functions (cf. [[Convex function (of a complex variable)|Convex function (of a complex variable)]]; [[Star-like function|Star-like function]]; [[Typically-real function|Typically-real function]]). The [[Convexity radius|convexity radius]] and the radius of  "star-likeness"  were found for a number of classes (see [[Limit of star-likeness|Limit of star-likeness]]).
  
 
The basic methods of the theory of univalent functions and some of the results obtained from them are given below.
 
The basic methods of the theory of univalent functions and some of the results obtained from them are given below.
Line 22: Line 97:
 
This method enables one to solve many problems in the theory of functions quite simply, in particular extremal problems in classes of functions that can be represented by means of Stieltjes integrals: convex functions, close-to-convex functions, star-like functions, typically-real functions, and functions with positive real part (see [[Carathéodory class|Carathéodory class]]). A variational method (see [[#References|[1]]]) was developed for classes of functions representable by Stieltjes integrals, by means of which a number of extremal problems have been solved. The method of internal variations (cf. [[Internal variations, method of|Internal variations, method of]]) was developed for such classes.
 
This method enables one to solve many problems in the theory of functions quite simply, in particular extremal problems in classes of functions that can be represented by means of Stieltjes integrals: convex functions, close-to-convex functions, star-like functions, typically-real functions, and functions with positive real part (see [[Carathéodory class|Carathéodory class]]). A variational method (see [[#References|[1]]]) was developed for classes of functions representable by Stieltjes integrals, by means of which a number of extremal problems have been solved. The method of internal variations (cf. [[Internal variations, method of|Internal variations, method of]]) was developed for such classes.
  
The convex hulls of certain subclasses of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562065.png" /> have been found (see [[#References|[3]]]). In particular, it has been proved that for every star-like function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562066.png" /> there exists a non-decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562069.png" /> and
+
The convex hulls of certain subclasses of $  S $
 +
have been found (see [[#References|[3]]]). In particular, it has been proved that for every star-like function $  f $
 +
there exists a non-decreasing function $  \mu $
 +
on $  [ 0 , 2 \pi ] $
 +
such that $  \mu ( 2 \pi ) - \mu ( 0) = 1 $
 +
and
 +
 
 +
$$
 +
f ( z)  = \
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562070.png" /></td> </tr></table>
+
\frac{z}{( 1 - e ^ {i \theta } z )  ^ {2} }
 +
  d \mu ( \theta ) .
 +
$$
  
 
See also [[Integral representation of an analytic function|Integral representation of an analytic function]]; [[Parametric representation of univalent functions|Parametric representation of univalent functions]]; [[Parametric representation method|Parametric representation method]].
 
See also [[Integral representation of an analytic function|Integral representation of an analytic function]]; [[Parametric representation of univalent functions|Parametric representation of univalent functions]]; [[Parametric representation method|Parametric representation method]].
  
 
==2. The method of boundary integration.==
 
==2. The method of boundary integration.==
With the aid of this method it has been proved, in particular, that an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562071.png" /> satisfies the inequality (see [[#References|[1]]])
+
With the aid of this method it has been proved, in particular, that an $  f \in S $
 +
satisfies the inequality (see [[#References|[1]]])
 +
 
 +
$$
 +
\left |
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562072.png" /></td> </tr></table>
+
\frac{z}{f(}
 +
z) + c _ {2} z + 1 - | z |  ^ {2} -
 +
2
 +
\frac{\mathbf E ( | z | ) }{\mathbf K ( | z | ) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562073.png" /></td> </tr></table>
+
\right | \leq
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562075.png" /> are complete elliptic integrals (cf. [[Elliptic integral|Elliptic integral]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562076.png" /> is fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562077.png" />, then this inequality determines the range of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562078.png" /> on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562079.png" />. Stronger versions of the distortion theorems were obtained, and theorems were proved on the distortion of chords in the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562081.png" /> (see [[Distortion theorems|Distortion theorems]] and [[#References|[1]]]).
+
$$
 +
\leq  \
 +
2 \left ( 1 -
 +
\frac{\mathbf E ( | z | ) }{
 +
\mathbf K ( | z | ) }
 +
\right ) ,\  | z | < 1 ,
 +
$$
 +
 
 +
where $  \mathbf E $
 +
and $  \mathbf K $
 +
are complete elliptic integrals (cf. [[Elliptic integral|Elliptic integral]]). If $  z $
 +
is fixed $  ( 0 < | z | < 1 ) $,  
 +
then this inequality determines the range of values of the functional $  c _ {2} z + z / f ( z) $
 +
on the class $  S $.  
 +
Stronger versions of the distortion theorems were obtained, and theorems were proved on the distortion of chords in the classes $  \Sigma $
 +
and $  \Sigma ( B) $(
 +
see [[Distortion theorems|Distortion theorems]] and [[#References|[1]]]).
  
 
See also [[Method of boundary integration|Method of boundary integration]]; [[Area principle|Area principle]].
 
See also [[Method of boundary integration|Method of boundary integration]]; [[Area principle|Area principle]].
  
 
==3. The area method.==
 
==3. The area method.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562082.png" /> be the class of systems of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562083.png" /> mapping the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562084.png" /> conformally and univalently onto pairwise disjoint (non-overlapping) domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562085.png" /> and normalized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562086.png" />. The following results have been obtained by means of the area theorem in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562087.png" />:
+
Let $  \mathfrak M ( a _ {1} \dots a _ {n} ) $
 +
be the class of systems of functions $  \{ f _ {k} ( x) \} _ {k=} 1  ^ {n} $
 +
mapping the disc $  \Delta = \{ | z | < 1 \} $
 +
conformally and univalently onto pairwise disjoint (non-overlapping) domains $  B _ {k} \ni a _ {k} $
 +
and normalized by the conditions $  f _ {k} ( 0) = a _ {k} $.  
 +
The following results have been obtained by means of the area theorem in the class $  \mathfrak M ( \infty , a _ {1} \dots a _ {n} ) $:
  
 
1) If
 
1) If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562088.png" /></td> </tr></table>
+
$$
 +
\{ f _ {k} \} _ {k=} 1  ^ {n}
 +
\in  \mathfrak M ( a _ {1} \dots a _ {n} ) ,\  a _ {k} \neq \infty ,
 +
$$
 +
 
 +
then
 +
 
 +
$$ \tag{3 }
 +
\prod _ { k= } 1 ^ { n }
 +
| f _ {k} ^ { \prime } ( 0) | ^ {| \gamma _ {k} |  ^ {2} }  \leq  \
 +
\prod _ {1 \leq  k < l \leq  n } | a _ {k} - a _ {l} | ^ {- 2  \mathop{\rm Re} ( \gamma _ {k} , \overline \gamma \; _ {l} ) } ,
 +
$$
 +
 
 +
$$
 +
\sum _ { k= } 1 ^ { n }  \gamma _ {k}  = 0 ;
 +
$$
 +
 
 +
this inequality generalizes an inequality previously known for real  $  \gamma _ {k} $
 +
to the class of complex  $  \gamma _ {k} $.
  
 +
2) If  $  \{ f _ {0} , f _ {1} \} \in \mathfrak M ( 0 , \infty ) $,
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{4 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562090.png" /></td> </tr></table>
+
\frac{1}{2 \pi }
  
this inequality generalizes an inequality previously known for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562091.png" /> to the class of complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562092.png" />.
+
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
| f _ {0} ( e  ^ {it} ) |  ^ {2}  d t  \cdot \
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562093.png" />, then
+
\frac{1}{2 \pi }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562094.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
| f _ {1} ( e  ^ {it} ) |  ^ {-} 2  d t  \leq  1 .
 +
$$
  
 
For the [[Bieberbach–Eilenberg functions|Bieberbach–Eilenberg functions]]
 
For the [[Bieberbach–Eilenberg functions|Bieberbach–Eilenberg functions]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562095.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \sum _ { k= } 1 ^  \infty  a _ {k} z  ^ {k}
 +
$$
  
 
there follows the inequality
 
there follows the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562096.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4prm)</td></tr></table>
+
$$ \tag{4'}
 +
\sum _ { k= } 1 ^  \infty  | a _ {k} |  ^ {2}  = \
  
and conditions have been determined for equality to hold in (4) and (4prm).
+
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
| f ( e  ^ {it} ) |  ^ {2}  d t  \leq  1 ;
 +
$$
 +
 
 +
and conditions have been determined for equality to hold in (4) and (4'}).
  
 
Using the area theorem for non-overlapping domains, bounds have been obtained for the approximation to a regular function on a closed multiply-connected domain by a rational function interpolating the given function at nodes uniformly distributed on the boundary of the domain (see [[#References|[4]]]). The range of values of the Schwarzian
 
Using the area theorem for non-overlapping domains, bounds have been obtained for the approximation to a regular function on a closed multiply-connected domain by a rational function interpolating the given function at nodes uniformly distributed on the boundary of the domain (see [[#References|[4]]]). The range of values of the Schwarzian
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562097.png" /></td> </tr></table>
+
$$
 +
\{ F ( z) , z \}  = \
 +
\left (
 +
\frac{F ^ { \prime\prime } ( z) }{F ^ { \prime } ( z) }
 +
\right )  ^  \prime  -
  
has been obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562098.png" />, and a number of other ranges of values have been found for classes of functions given on multiply-connected domains (see [[#References|[4]]], [[#References|[5]]]).
+
\frac{1}{2}
 +
 
 +
\left (
 +
\frac{F ^ { \prime\prime } ( z) }{F ^ { \prime } ( z) }
 +
\right )  ^ {2}
 +
$$
 +
 
 +
has been obtained for $  F \in \Sigma ( B) $,  
 +
and a number of other ranges of values have been found for classes of functions given on multiply-connected domains (see [[#References|[4]]], [[#References|[5]]]).
  
 
==4. Löwner's method.==
 
==4. Löwner's method.==
K. Löwner himself (1923) found the exact bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u09562099.png" /> for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620100.png" /> and exact bounds for the coefficients of the expansion of the function inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620101.png" />, in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620102.png" />. In particular, an exact form of the rotation theorem in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620103.png" /> was obtained by this method (see [[Rotation theorems|Rotation theorems]]). The following theorem was proved: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620104.png" /> and given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620106.png" />, the following inequality is valid:
+
K. Löwner himself (1923) found the exact bound $  | c _ {3} | \leq  3 $
 +
for functions $  f \in S $
 +
and exact bounds for the coefficients of the expansion of the function inverse to $  f $,  
 +
in a neighbourhood of the point $  w = 0 $.  
 +
In particular, an exact form of the rotation theorem in the class $  S $
 +
was obtained by this method (see [[Rotation theorems|Rotation theorems]]). The following theorem was proved: For $  f \in S $
 +
and given $  z \in \Delta $
 +
and $  | f ( z) | $,  
 +
the following inequality is valid:
 +
 
 +
$$ \tag{5 }
 +
| f ^ { \prime } ( z) |
 +
\leq 
 +
\frac{1}{1 - | z |  ^ {2} }
 +
 
 +
\left |
 +
\frac{f ( z) }{z}
 +
\right |  ^ {2}
 +
( 1 - x  ^ {2} )  ^ {2}
 +
\left |
 +
\frac{x}{2}
 +
\right | ^ {4 x  ^ {2} / ( 1 - x  ^ {2} ) } ,
 +
$$
 +
 
 +
where  $  x $,
 +
$  | x | < | z | $,
 +
is determined by the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$
 +
\left |
 +
\frac{f ( x) }{z}
 +
\right |
 +
( 1 + x ) ^ {2}
 +
\left |
 +
\frac{z}{x}
 +
\right | ^ {2 x / ( 1 + x ) }
 +
=  1 .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620109.png" />, is determined by the condition
+
Inequality (5) is sharp; it implies the following sharp inequalities in the class  $  S $(
 +
$  0 \leq  \theta < 2 \pi $,
 +
0 \leq  r < 1 $):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620110.png" /></td> </tr></table>
+
$$ \tag{6 }
 +
\left . \begin{array}{c}
  
Inequality (5) is sharp; it implies the following sharp inequalities in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620111.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620113.png" />):
+
| f ( r e ^ {i \theta } ) | + | f ( - r e ^ {i \theta } ) |  \leq  \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620114.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
\frac{r}{( 1 - r )  ^ {2} }
 +
+
 +
 
 +
\frac{r}{( 1 + r )  ^ {2} }
 +
,
 +
\\
 +
 
 +
| f ^ { \prime } ( r e ^
 +
{i \theta } ) | + | f ^ { \prime } ( - r e ^ {i \theta } ) |  \leq  \
 +
1+
 +
\frac{r}{( 1 - r )  ^ {3} }
 +
+
 +
1-  
 +
\frac{r}{( 1 + r ) ^ {3} }
 +
.
 +
 +
\end{array}
 +
\right \}
 +
$$
  
 
By means of distortion theorems it has been established that the Koebe function
 
By means of distortion theorems it has been established that the Koebe function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620115.png" /></td> </tr></table>
+
$$
 +
K _  \alpha  ( z)  = z
 +
( 1 - e ^ {i \alpha } z )  ^ {-} 2  \in  S
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620116.png" /> real) realizes the maximum of the linear measure of covering the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620117.png" /> by the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620118.png" /> of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620119.png" /> under mappings by functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620120.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620121.png" />. This property of functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620122.png" /> implies bounds for the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620123.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620124.png" />, bounds for the average modulus of a function, and other bounds in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620125.png" />; these are asymptotically sharp as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620126.png" /> (see [[#References|[1]]]).
+
( $  \alpha $
 +
real) realizes the maximum of the linear measure of covering the circle $  | w | = \rho $
 +
by the image $  B ( r) $
 +
of the disc $  \Delta _ {r} = \{ | z | < r < 1 \} $
 +
under mappings by functions of class $  S $
 +
when  $  \rho > e ^ {\pi / e } r $.  
 +
This property of functions of class $  S $
 +
implies bounds for the area $  \sigma ( r) $
 +
of the domain $  B ( r) $,  
 +
bounds for the average modulus of a function, and other bounds in the class $  S $;  
 +
these are asymptotically sharp as $  r \rightarrow 1 $(
 +
see [[#References|[1]]]).
  
A convenient reduction of extremal problems on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620127.png" /> and some of its subclasses to certain extremal problems on a simpler class has been proposed (see [[Carathéodory class|Carathéodory class]]). This turns out to be applicable to the solution of several extremal problems, in particular to finding the range of values of the system of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620128.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620130.png" />, is fixed) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620131.png" /> (see [[#References|[6]]]).
+
A convenient reduction of extremal problems on $  S $
 +
and some of its subclasses to certain extremal problems on a simpler class has been proposed (see [[Carathéodory class|Carathéodory class]]). This turns out to be applicable to the solution of several extremal problems, in particular to finding the range of values of the system of functionals $  \{  \mathop{\rm ln} ( f ( z) / z),  \mathop{\rm ln}  f ^ { \prime } ( z) \} $(
 +
here $  z $,
 +
$  0 < | z | < 1 $,  
 +
is fixed) for $  f \in S $(
 +
see [[#References|[6]]]).
  
Löwner's method has been successfully applied to investigate the properties of level curves and to solve extremal problems in the subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620132.png" /> of bounded functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620133.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620135.png" /> (see [[#References|[6]]]).
+
Löwner's method has been successfully applied to investigate the properties of level curves and to solve extremal problems in the subclass $  S _ {M} $
 +
of bounded functions $  f \in S $:  
 +
$  | f ( z) | \leq  M $,  
 +
$  z \in \Delta $(
 +
see [[#References|[6]]]).
  
 
See also [[Löwner equation|Löwner equation]]; [[Löwner method|Löwner method]]; [[Parametric representation method|Parametric representation method]].
 
See also [[Löwner equation|Löwner equation]]; [[Löwner method|Löwner method]]; [[Parametric representation method|Parametric representation method]].
  
 
==5. Variational methods.==
 
==5. Variational methods.==
Boundary and internal variations in the solution of extremal problems lead to differential equations for the boundaries of the extremal domains and for extremal functions, respectively. As a rule, the left-hand side of these equations is a [[Quadratic differential|quadratic differential]]. Various qualitative characteristics of the functions realizing the extremum can be obtained by investigating the properties of the corresponding quadratic differentials. In particular, it turns out that for a large number of extremal problems in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620136.png" /> (and in other classes), the extremal function maps the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620137.png" /> onto the whole plane with a finite number of analytic slits. Sometimes the differential equation for the extremal function can be integrated, and one obtains the extremal quantity and all extremal functions in the problem considered. More often one only obtains one or a few equations for the extremal quantity. Some results obtained by variational methods are listed below.
+
Boundary and internal variations in the solution of extremal problems lead to differential equations for the boundaries of the extremal domains and for extremal functions, respectively. As a rule, the left-hand side of these equations is a [[Quadratic differential|quadratic differential]]. Various qualitative characteristics of the functions realizing the extremum can be obtained by investigating the properties of the corresponding quadratic differentials. In particular, it turns out that for a large number of extremal problems in the class $  S $(
 +
and in other classes), the extremal function maps the disc $  \Delta $
 +
onto the whole plane with a finite number of analytic slits. Sometimes the differential equation for the extremal function can be integrated, and one obtains the extremal quantity and all extremal functions in the problem considered. More often one only obtains one or a few equations for the extremal quantity. Some results obtained by variational methods are listed below.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620138.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620141.png" />, do not belong to the image of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620142.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620143.png" />, and that
+
Suppose that $  F \in \Sigma $,  
 +
that $  w _ {k} $,  
 +
$  k = 1 \dots n $,  
 +
$  n \geq  2 $,  
 +
do not belong to the image of the domain $  \Delta  ^  \prime  = \{ | z | > 1 \} $
 +
under the mapping $  w = F ( z) $,  
 +
and that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620144.png" /></td> </tr></table>
+
$$
 +
d _ {n} ( F  )  = \
 +
\prod _ {1 \leq  k < l \leq  n }
 +
| w _ {k} - w _ {l} | .
 +
$$
  
 
It has been proved that
 
It has been proved that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620145.png" /></td> </tr></table>
+
$$
 +
d _ {3} ( F  )  = \
 +
| ( w _ {1} - w _ {2} ) ( w _ {2} - w _ {3} ) ( w _ {3} - w _ {1} ) |
 +
\leq  12 \sqrt 3 ,
 +
$$
  
 
with equality only for
 
with equality only for
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620146.png" /></td> </tr></table>
+
$$
 +
F ( z)  = z
 +
( 1 + e ^ {i \alpha } z  ^ {-} 3 )  ^ {2/3} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620147.png" /> is real (see [[#References|[1]]]).
+
where $  \alpha $
 +
is real (see [[#References|[1]]]).
  
 
It has been proved (see [[#References|[1]]]) that the range of values of the functional
 
It has been proved (see [[#References|[1]]]) that the range of values of the functional
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620148.png" /></td> </tr></table>
+
$$
 +
= \sum _ {\nu , \nu  ^  \prime  = 1 } ^ { n }
 +
\gamma _  \nu  \gamma _ {\nu  ^  \prime  }  \mathop{\rm ln} \
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620149.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620150.png" /> are given numbers not all zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620151.png" /> are given points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620152.png" />, is the disc
+
\frac{F ( \zeta _  \nu  ) - F ( \zeta _ {\nu  ^  \prime  } ) }{\zeta _  \nu  - \zeta _ {\nu  ^  \prime  } }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620153.png" /></td> </tr></table>
+
$$
  
The problem has been investigated of extremizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620155.png" />, in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620156.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620157.png" /> that are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620158.png" /> and do not take on given values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620159.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620160.png" /> (see [[#References|[1]]]). The special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620161.png" /> is the problem of determining a continuum of least capacity (for a consideration of this problem and its generalizations see [[#References|[7]]]).
+
for  $  F \in \Sigma $,
 +
where  $  \gamma _  \nu  $
 +
are given numbers not all zero and  $  \zeta _  \nu  $
 +
are given points in  $  \Delta  ^  \prime  $,
 +
is the disc
 +
 
 +
$$
 +
| w |  \leq  -  \mathop{\rm Re} \
 +
\sum _ {\nu , \nu  ^  \prime  = 1 } ^ { n }
 +
\gamma _  \nu  \overline \gamma \; _ {\nu  ^  \prime  }  \mathop{\rm ln}
 +
( 1 - \zeta _  \nu  ^ {-} 1 {\zeta _ {\nu  ^  \prime  }  ^ {-} 1 } bar ) .
 +
$$
 +
 
 +
The problem has been investigated of extremizing $  | c _ {n} | $,  
 +
$  n \geq  1 $,  
 +
in the class $  S _ {a} $
 +
of functions $  f ( z) = c _ {1} z + c _ {2} z  ^ {2} + \dots $
 +
that are regular and univalent in the disc $  \Delta $
 +
and do not take on given values $  a _ {1} \dots a _ {n} $
 +
in $  \Delta $(
 +
see [[#References|[1]]]). The special case $  n = 1 $
 +
is the problem of determining a continuum of least capacity (for a consideration of this problem and its generalizations see [[#References|[7]]]).
  
 
Various problems for non-overlapping domains have been investigated by a variational method. Thus, the problem of maximizing the product
 
Various problems for non-overlapping domains have been investigated by a variational method. Thus, the problem of maximizing the product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620162.png" /></td> </tr></table>
+
$$
 +
I _ {n}  = \
 +
\prod _ { k= } 1 ^ { n }
 +
| f _ {k} ^ { \prime } ( 0) |
 +
$$
  
in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620163.png" /> has been considered (see [[#References|[1]]]). A precise bound has been obtained for the product
+
in the class $  \mathfrak M ( a _ {1} \dots a _ {n} ) $
 +
has been considered (see [[#References|[1]]]). A precise bound has been obtained for the product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620164.png" /></td> </tr></table>
+
$$
 +
\prod _ { k= } 1 ^ { n }
 +
| f _ {k} ^ { \prime } ( 0) | ^ {\alpha _ {k} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620165.png" /> are any given positive numbers, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620166.png" /> and 3 (see [[#References|[1]]]). This problem is equivalent to the problem of finding the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620167.png" /> of the system of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620168.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620169.png" />.
+
where $  \alpha _ {k} $
 +
are any given positive numbers, for $  n = 2 $
 +
and 3 (see [[#References|[1]]]). This problem is equivalent to the problem of finding the range $  D $
 +
of the system of functionals $  ( | f _ {1} ( 0) | \dots | f _ {n} ( 0) | ) $
 +
in the class $  \mathfrak M ( a _ {1} \dots a _ {n} ) $.
  
 
See also [[Variational principles (in complex function theory)|Variational principles (in complex function theory)]]; [[Variation of a univalent function|Variation of a univalent function]]; [[Internal variations, method of|Internal variations, method of]]; [[Boundary variation, method of|Boundary variation, method of]]; [[Variation-parametric method|Variation-parametric method]].
 
See also [[Variational principles (in complex function theory)|Variational principles (in complex function theory)]]; [[Variation of a univalent function|Variation of a univalent function]]; [[Internal variations, method of|Internal variations, method of]]; [[Boundary variation, method of|Boundary variation, method of]]; [[Variation-parametric method|Variation-parametric method]].
  
 
==6. The method of the extremal metric.==
 
==6. The method of the extremal metric.==
In the solution of extremal problems by the method of the extremal metric, a fundamental role is played, as a rule, by the metric generated by a certain quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620170.png" />. This is the same quadratic differential that arises in the solution of the problem by the variational method. As an example, two results obtained by this method are given below (see [[#References|[1]]], [[#References|[7]]]–[[#References|[9]]]).
+
In the solution of extremal problems by the method of the extremal metric, a fundamental role is played, as a rule, by the metric generated by a certain quadratic differential $  Q ( z)  d z  ^ {2} $.  
 +
This is the same quadratic differential that arises in the solution of the problem by the variational method. As an example, two results obtained by this method are given below (see [[#References|[1]]], [[#References|[7]]]–[[#References|[9]]]).
  
By means of the general coefficient theorem, J.A. Jenkins (1960) has solved the problem of the range of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620171.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620172.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620173.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620174.png" /> of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620175.png" /> with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620176.png" />. In the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620178.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620179.png" /> is the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620182.png" />, that are meromorphic and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620183.png" />, he clarified the influence of the vanishing of a certain number of the initial coefficients on the growth of the subsequent ones.
+
By means of the general coefficient theorem, J.A. Jenkins (1960) has solved the problem of the range of values of the functional $  f ( z) $
 +
for fixed $  z $
 +
in the disc $  \Delta = \{ | z | < 1 \} $
 +
in the class $  S _ {r} $
 +
of functions in $  S $
 +
with real coefficients $  c _ {2} , c _ {3} ,\dots $.  
 +
In the classes $  \Sigma $
 +
and $  M $,  
 +
where $  M $
 +
is the class of functions $  f $,
 +
$  f ( 0) = 0 $,  
 +
$  f ^ { \prime } ( 0) = 1 $,  
 +
that are meromorphic and univalent in the disc $  \Delta $,  
 +
he clarified the influence of the vanishing of a certain number of the initial coefficients on the growth of the subsequent ones.
  
A supplement to the general coefficient theorem has been given in the case when the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620184.png" /> has no poles of order higher than one; in addition, by means of the extremal-metric approach, very general theorems have been established on the covering of curves under a univalent conformal mapping of simply- and doubly-connected domains, including, in particular, a refinement of the result on covering of intervals for functions meromorphic and univalent on the disc, and an analogous result for a circular annulus (see [[#References|[1]]]).
+
A supplement to the general coefficient theorem has been given in the case when the differential $  Q ( z)  d z  ^ {2} $
 +
has no poles of order higher than one; in addition, by means of the extremal-metric approach, very general theorems have been established on the covering of curves under a univalent conformal mapping of simply- and doubly-connected domains, including, in particular, a refinement of the result on covering of intervals for functions meromorphic and univalent on the disc, and an analogous result for a circular annulus (see [[#References|[1]]]).
  
 
See also [[Grötzsch principle|Grötzsch principle]]; [[Grötzsch theorems|Grötzsch theorems]]; [[Strip method (analytic functions)|Strip method (analytic functions)]]; [[Quadratic differential|Quadratic differential]]; [[Bieberbach–Eilenberg functions|Bieberbach–Eilenberg functions]]; [[Extremal metric, method of the|Extremal metric, method of the]].
 
See also [[Grötzsch principle|Grötzsch principle]]; [[Grötzsch theorems|Grötzsch theorems]]; [[Strip method (analytic functions)|Strip method (analytic functions)]]; [[Quadratic differential|Quadratic differential]]; [[Bieberbach–Eilenberg functions|Bieberbach–Eilenberg functions]]; [[Extremal metric, method of the|Extremal metric, method of the]].
  
 
==7. The method of symmetrization.==
 
==7. The method of symmetrization.==
Several complicated extremal problems not lending themselves to solution by other methods have been solved by this method, often in conjunction with others. For example, the following problems are of this kind (see [[#References|[1]]], [[#References|[7]]]–[[#References|[10]]]). For functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620185.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620186.png" />, a sharp upper bound has been found for the set of points of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620188.png" />, not belonging to the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620189.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620190.png" />. In conjunction with the method of the extremal metric, a sharp upper bound has been found for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620191.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620193.png" />, for
+
Several complicated extremal problems not lending themselves to solution by other methods have been solved by this method, often in conjunction with others. For example, the following problems are of this kind (see [[#References|[1]]], [[#References|[7]]]–[[#References|[10]]]). For functions $  f $
 +
in the class $  S $,  
 +
a sharp upper bound has been found for the set of points of the circle $  | w | = R $,
 +
$  1 / 4 \leq  R < 1 $,  
 +
not belonging to the image of the disc $  \Delta $
 +
under the mapping $  w = f ( z) $.  
 +
In conjunction with the method of the extremal metric, a sharp upper bound has been found for $  | f ( z) | $
 +
for fixed $  | z | = r $,
 +
0 < r < 1 $,  
 +
for
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620194.png" /></td> </tr></table>
+
$$
 +
f ( z)  = z + c _ {2} z  ^ {2} + \dots  \in  S
 +
$$
  
with given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620195.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620196.png" />; the inequalities (6) have been generalized and extended to the class of functions that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620197.png" />-valent in the mean on the circle:
+
with given $  c _ {2} = c = \textrm{ const } $,  
 +
0 \leq  c \leq  2 $;  
 +
the inequalities (6) have been generalized and extended to the class of functions that are $  p $-
 +
valent in the mean on the circle:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620198.png" /></td> </tr></table>
+
$$
 +
f ( z)  = z  ^ {p} + c _ {p+} 1 z  ^ {p+} 1 + \dots
 +
$$
  
 
(see [[Multivalent function|Multivalent function]]).
 
(see [[Multivalent function|Multivalent function]]).
  
By the method of symmetrization it has been proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620199.png" /> is a convex and non-decreasing function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620200.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620202.png" />,
+
By the method of symmetrization it has been proved that if $  \phi $
 +
is a convex and non-decreasing function on $  ( - \infty , + \infty ) $,  
 +
then for $  f \in S $
 +
and  $  0 < r < 1 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620203.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {- \pi } ^  \pi 
 +
\Phi (  \mathop{\rm ln}  | f ( r e  ^ {it} ) | )  d t
 +
\leq  \int\limits _ {- \pi } ^  \pi 
 +
\Phi (  \mathop{\rm ln}  | K ( r e  ^ {it} ) | )  d t ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620204.png" /> (see [[#References|[11]]]). If equality holds for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620206.png" />, and for some strictly convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620207.png" />, then
+
where $  K ( z) = z ( 1 - z )  ^ {-} 2 $(
 +
see [[#References|[11]]]). If equality holds for some $  r $,
 +
$  0 < r < 1 $,  
 +
and for some strictly convex function $  \phi $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620208.png" /></td> </tr></table>
+
$$
 +
f ( z)  = e ^ {i \alpha } K ( e ^ {i \alpha } z ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095620/u095620209.png" /> is real.
+
where $  \alpha $
 +
is real.
  
 
See [[#References|[12]]], [[#References|[13]]] for applications of the method of symmetrization to multiply-connected domains. See also [[Symmetrization method|Symmetrization method]].
 
See [[#References|[12]]], [[#References|[13]]] for applications of the method of symmetrization to multiply-connected domains. See also [[Symmetrization method|Symmetrization method]].
Line 172: Line 509:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Lebedev,  I.A. Aleksandrov,  "On the variational method in classes of functions representable by means of Stieltjes integrals"  ''Proc. Steklov Inst. Math.'' , '''94'''  (1969)  pp. 91–104  ''Trudy Mat. Inst. Steklov.'' , '''94'''  (1968)  pp. 79–89</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Brickman,  T.H. MacGregor,  D.R. Wilken,  "Convex hulls of some classical families of univalent functions"  ''Trans. Amer. Math. Soc.'' , '''156'''  (1971)  pp. 91–107</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.A. Lebedev,  "The area principle in the theory of univalent functions" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Milin,  "Univalent functions and orthonormal systems" , ''Transl. Math. Monogr.'' , '''49''' , Amer. Math. Soc.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.A. Aleksandrov,  "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.V. Kuz'mina,  "Moduli of families of curves and quadratic differentials"  ''Proc. Steklov Inst. Math.'' , '''139'''  (1982)  ''Trudy Mat. Inst. Steklov.'' , '''139'''  (1980)  pp. 1–241</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  C. Pommerenke,  "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  W.K. Hayman,  "Multivalent functions" , Cambridge Univ. Press  (1958)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A. Baernstein,  "Integral means, univalent functions and circular symmetrization"  ''Acta Math.'' , '''133'''  (1974)  pp. 139–169</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  O.P. Mityuk,  "The symmetrization principle for multiply-connected domains and certain of its applications"  ''Ukrain. Mat. Zh.'' , '''17''' :  4  (1965)  pp. 46–54  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  I.P. Mityuk,  "The symmetrization principle for an annulus and certain of its applications"  ''Sibirsk. Mat. Zh.'' , '''6''' :  6  (1965)  pp. 1282–1291  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Lebedev,  I.A. Aleksandrov,  "On the variational method in classes of functions representable by means of Stieltjes integrals"  ''Proc. Steklov Inst. Math.'' , '''94'''  (1969)  pp. 91–104  ''Trudy Mat. Inst. Steklov.'' , '''94'''  (1968)  pp. 79–89</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Brickman,  T.H. MacGregor,  D.R. Wilken,  "Convex hulls of some classical families of univalent functions"  ''Trans. Amer. Math. Soc.'' , '''156'''  (1971)  pp. 91–107</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.A. Lebedev,  "The area principle in the theory of univalent functions" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Milin,  "Univalent functions and orthonormal systems" , ''Transl. Math. Monogr.'' , '''49''' , Amer. Math. Soc.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.A. Aleksandrov,  "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.V. Kuz'mina,  "Moduli of families of curves and quadratic differentials"  ''Proc. Steklov Inst. Math.'' , '''139'''  (1982)  ''Trudy Mat. Inst. Steklov.'' , '''139'''  (1980)  pp. 1–241</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  C. Pommerenke,  "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  W.K. Hayman,  "Multivalent functions" , Cambridge Univ. Press  (1958)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A. Baernstein,  "Integral means, univalent functions and circular symmetrization"  ''Acta Math.'' , '''133'''  (1974)  pp. 139–169</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  O.P. Mityuk,  "The symmetrization principle for multiply-connected domains and certain of its applications"  ''Ukrain. Mat. Zh.'' , '''17''' :  4  (1965)  pp. 46–54  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  I.P. Mityuk,  "The symmetrization principle for an annulus and certain of its applications"  ''Sibirsk. Mat. Zh.'' , '''6''' :  6  (1965)  pp. 1282–1291  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:27, 6 June 2020


A regular or meromorphic function $ f $ in a domain $ B $ of the extended complex plane $ \overline{\mathbf C}\; $ such that $ f ( z _ {1} ) \neq f ( z _ {2} ) $ whenever $ z _ {1} \neq z _ {2} $, $ z _ {1} , z _ {2} \in B $, that is, $ f $ is a one-to-one mapping from $ B $ into $ \overline{\mathbf C}\; $. The inverse function $ z = f ^ { - 1 } ( w) $ is then also univalent. Multivalent functions (cf. Multivalent function), and in particular $ p $- valent functions, are a generalization of univalent functions.

In the study of univalent functions one of the fundamental problems is whether there exists a univalent mapping from a given domain $ B $ onto a given domain $ B ^ \prime $. A necessary condition for the existence of such a mapping is that $ B $ and $ B ^ \prime $ have equal degrees of connectivity (see, for example, [1]). If $ B $ and $ B ^ \prime $ are simply-connected domains whose boundaries contain more than one point, then this condition is also sufficient (see Riemann theorem), and the problem reduces to mapping a given domain onto a disc. In this connection, a special role is played in the theory of univalent functions on simply-connected domains by the class $ S $ of functions $ f $ that are regular and univalent on the disc $ \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $, normalized by the conditions $ f ( 0) = 0 $, $ f ^ { \prime } ( 0) = 1 $, and having the expansion

$$ \tag{1 } f ( z) = z + c _ {2} z ^ {2} + \dots + c _ {n} z ^ {n} + \dots ,\ \ z \in \Delta . $$

In the case of multiply-connected domains, mappings of a given multiply-connected domain onto so-called canonical domains are studied (see Conformal mapping). Let $ \Sigma ( B) $ be the class of functions $ F $ that are meromorphic and univalent on a domain $ B $ containing the point $ \infty $, and having an expansion

$$ \tag{2 } F ( z) = z + \alpha _ {0} + \alpha _ {1} z ^ {-} 1 + \dots + \alpha _ {n} z ^ {-} n + \dots $$

in a neighbourhood of $ \infty $. If $ B = \{ {z \in \overline{\mathbf C}\; } : {| z | > 1 } \} = \Delta ^ \prime $, then this class is denoted by $ \Sigma $.

The basic problems in the theory of univalent functions are the following: 1) the study of the correspondence of boundaries under conformal mapping (see Boundary correspondence (under conformal mapping); Limit elements; Attainable boundary point); 2) obtaining univalency conditions; and 3) the solution of various extremal problems in function theory, in particular obtaining bounds for various functionals and for the range of values of functionals (see below) and systems of them in some class or other.

Suppose that $ K $ is some class (set) of regular or meromorphic functions, and suppose that a complex functional $ w = \phi ( f ) $( or system of functionals $ \{ \phi _ {k} ( f ) \} _ {k=} 1 ^ {n} $) is given on $ K $. The range of values of the functional $ \phi ( f ) $( or of the system of functionals $ \{ \phi _ {k} ( f ) \} _ {k=} 1 ^ {n} $) on the class $ K $ is the set $ D $ of points $ w = \phi ( f ) $ in $ \mathbf C $( respectively, the set in points $ ( \phi _ {1} ( f ) \dots \phi _ {n} ( f )) $ in $ n $- dimensional complex space $ \mathbf C ^ {n} $) such that $ f \in K $. Real-valued functionals are also considered. Any set $ D ^ \prime $ containing $ D $ is called a majorant domain of the functional (or of the system of functionals). Knowledge of the range of values of a functional enables one to reduce the solution of a number of extremal problems to simple problems in analysis. For example, if the range of values $ D $ is known for the functional $ f ( z _ {0} ) $, $ f \in K $( $ z _ {0} $ fixed), then the problem of finding upper and lower bounds for $ | f ( z _ {0} ) | $ reduces to finding the points of $ D $ farthest from and closest to the point $ w = 0 $.

The first substantial results in the theory of univalent functions were obtained using the area principle. With the aid of the outer area theorem (1916), L. Bieberbach obtained precise upper and lower bounds for $ | f ( z) | $ and $ | f ^ { \prime } ( z) | $ for $ f \in S $( see Distortion theorems), gave the bound $ | c _ {2} | \leq 2 $ for $ f \in S $ and conjectured that $ | c _ {n} | \leq n $ for $ f \in S $( see Bieberbach conjecture; Coefficient problem). He also found the exact value of the Koebe constant. Bounds were found for the modulus of a function and its derivative, as well as other bounds for the classes of convex, star-like, typically-real, etc., functions (cf. Convex function (of a complex variable); Star-like function; Typically-real function). The convexity radius and the radius of "star-likeness" were found for a number of classes (see Limit of star-likeness).

The basic methods of the theory of univalent functions and some of the results obtained from them are given below.

1. The method of integral representations.

This method enables one to solve many problems in the theory of functions quite simply, in particular extremal problems in classes of functions that can be represented by means of Stieltjes integrals: convex functions, close-to-convex functions, star-like functions, typically-real functions, and functions with positive real part (see Carathéodory class). A variational method (see [1]) was developed for classes of functions representable by Stieltjes integrals, by means of which a number of extremal problems have been solved. The method of internal variations (cf. Internal variations, method of) was developed for such classes.

The convex hulls of certain subclasses of $ S $ have been found (see [3]). In particular, it has been proved that for every star-like function $ f $ there exists a non-decreasing function $ \mu $ on $ [ 0 , 2 \pi ] $ such that $ \mu ( 2 \pi ) - \mu ( 0) = 1 $ and

$$ f ( z) = \ \int\limits _ { 0 } ^ { {2 } \pi } \frac{z}{( 1 - e ^ {i \theta } z ) ^ {2} } d \mu ( \theta ) . $$

See also Integral representation of an analytic function; Parametric representation of univalent functions; Parametric representation method.

2. The method of boundary integration.

With the aid of this method it has been proved, in particular, that an $ f \in S $ satisfies the inequality (see [1])

$$ \left | \frac{z}{f(} z) + c _ {2} z + 1 - | z | ^ {2} - 2 \frac{\mathbf E ( | z | ) }{\mathbf K ( | z | ) } \right | \leq $$

$$ \leq \ 2 \left ( 1 - \frac{\mathbf E ( | z | ) }{ \mathbf K ( | z | ) } \right ) ,\ | z | < 1 , $$

where $ \mathbf E $ and $ \mathbf K $ are complete elliptic integrals (cf. Elliptic integral). If $ z $ is fixed $ ( 0 < | z | < 1 ) $, then this inequality determines the range of values of the functional $ c _ {2} z + z / f ( z) $ on the class $ S $. Stronger versions of the distortion theorems were obtained, and theorems were proved on the distortion of chords in the classes $ \Sigma $ and $ \Sigma ( B) $( see Distortion theorems and [1]).

See also Method of boundary integration; Area principle.

3. The area method.

Let $ \mathfrak M ( a _ {1} \dots a _ {n} ) $ be the class of systems of functions $ \{ f _ {k} ( x) \} _ {k=} 1 ^ {n} $ mapping the disc $ \Delta = \{ | z | < 1 \} $ conformally and univalently onto pairwise disjoint (non-overlapping) domains $ B _ {k} \ni a _ {k} $ and normalized by the conditions $ f _ {k} ( 0) = a _ {k} $. The following results have been obtained by means of the area theorem in the class $ \mathfrak M ( \infty , a _ {1} \dots a _ {n} ) $:

1) If

$$ \{ f _ {k} \} _ {k=} 1 ^ {n} \in \mathfrak M ( a _ {1} \dots a _ {n} ) ,\ a _ {k} \neq \infty , $$

then

$$ \tag{3 } \prod _ { k= } 1 ^ { n } | f _ {k} ^ { \prime } ( 0) | ^ {| \gamma _ {k} | ^ {2} } \leq \ \prod _ {1 \leq k < l \leq n } | a _ {k} - a _ {l} | ^ {- 2 \mathop{\rm Re} ( \gamma _ {k} , \overline \gamma \; _ {l} ) } , $$

$$ \sum _ { k= } 1 ^ { n } \gamma _ {k} = 0 ; $$

this inequality generalizes an inequality previously known for real $ \gamma _ {k} $ to the class of complex $ \gamma _ {k} $.

2) If $ \{ f _ {0} , f _ {1} \} \in \mathfrak M ( 0 , \infty ) $, then

$$ \tag{4 } \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } | f _ {0} ( e ^ {it} ) | ^ {2} d t \cdot \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } | f _ {1} ( e ^ {it} ) | ^ {-} 2 d t \leq 1 . $$

For the Bieberbach–Eilenberg functions

$$ f ( z) = \sum _ { k= } 1 ^ \infty a _ {k} z ^ {k} $$

there follows the inequality

$$ \tag{4'} \sum _ { k= } 1 ^ \infty | a _ {k} | ^ {2} = \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } | f ( e ^ {it} ) | ^ {2} d t \leq 1 ; $$

and conditions have been determined for equality to hold in (4) and (4'}).

Using the area theorem for non-overlapping domains, bounds have been obtained for the approximation to a regular function on a closed multiply-connected domain by a rational function interpolating the given function at nodes uniformly distributed on the boundary of the domain (see [4]). The range of values of the Schwarzian

$$ \{ F ( z) , z \} = \ \left ( \frac{F ^ { \prime\prime } ( z) }{F ^ { \prime } ( z) } \right ) ^ \prime - \frac{1}{2} \left ( \frac{F ^ { \prime\prime } ( z) }{F ^ { \prime } ( z) } \right ) ^ {2} $$

has been obtained for $ F \in \Sigma ( B) $, and a number of other ranges of values have been found for classes of functions given on multiply-connected domains (see [4], [5]).

4. Löwner's method.

K. Löwner himself (1923) found the exact bound $ | c _ {3} | \leq 3 $ for functions $ f \in S $ and exact bounds for the coefficients of the expansion of the function inverse to $ f $, in a neighbourhood of the point $ w = 0 $. In particular, an exact form of the rotation theorem in the class $ S $ was obtained by this method (see Rotation theorems). The following theorem was proved: For $ f \in S $ and given $ z \in \Delta $ and $ | f ( z) | $, the following inequality is valid:

$$ \tag{5 } | f ^ { \prime } ( z) | \leq \frac{1}{1 - | z | ^ {2} } \left | \frac{f ( z) }{z} \right | ^ {2} ( 1 - x ^ {2} ) ^ {2} \left | \frac{x}{2} \right | ^ {4 x ^ {2} / ( 1 - x ^ {2} ) } , $$

where $ x $, $ | x | < | z | $, is determined by the condition

$$ \left | \frac{f ( x) }{z} \right | ( 1 + x ) ^ {2} \left | \frac{z}{x} \right | ^ {2 x / ( 1 + x ) } = 1 . $$

Inequality (5) is sharp; it implies the following sharp inequalities in the class $ S $( $ 0 \leq \theta < 2 \pi $, $ 0 \leq r < 1 $):

$$ \tag{6 } \left . \begin{array}{c} | f ( r e ^ {i \theta } ) | + | f ( - r e ^ {i \theta } ) | \leq \ \frac{r}{( 1 - r ) ^ {2} } + \frac{r}{( 1 + r ) ^ {2} } , \\ | f ^ { \prime } ( r e ^ {i \theta } ) | + | f ^ { \prime } ( - r e ^ {i \theta } ) | \leq \ 1+ \frac{r}{( 1 - r ) ^ {3} } + 1- \frac{r}{( 1 + r ) ^ {3} } . \end{array} \right \} $$

By means of distortion theorems it has been established that the Koebe function

$$ K _ \alpha ( z) = z ( 1 - e ^ {i \alpha } z ) ^ {-} 2 \in S $$

( $ \alpha $ real) realizes the maximum of the linear measure of covering the circle $ | w | = \rho $ by the image $ B ( r) $ of the disc $ \Delta _ {r} = \{ | z | < r < 1 \} $ under mappings by functions of class $ S $ when $ \rho > e ^ {\pi / e } r $. This property of functions of class $ S $ implies bounds for the area $ \sigma ( r) $ of the domain $ B ( r) $, bounds for the average modulus of a function, and other bounds in the class $ S $; these are asymptotically sharp as $ r \rightarrow 1 $( see [1]).

A convenient reduction of extremal problems on $ S $ and some of its subclasses to certain extremal problems on a simpler class has been proposed (see Carathéodory class). This turns out to be applicable to the solution of several extremal problems, in particular to finding the range of values of the system of functionals $ \{ \mathop{\rm ln} ( f ( z) / z), \mathop{\rm ln} f ^ { \prime } ( z) \} $( here $ z $, $ 0 < | z | < 1 $, is fixed) for $ f \in S $( see [6]).

Löwner's method has been successfully applied to investigate the properties of level curves and to solve extremal problems in the subclass $ S _ {M} $ of bounded functions $ f \in S $: $ | f ( z) | \leq M $, $ z \in \Delta $( see [6]).

See also Löwner equation; Löwner method; Parametric representation method.

5. Variational methods.

Boundary and internal variations in the solution of extremal problems lead to differential equations for the boundaries of the extremal domains and for extremal functions, respectively. As a rule, the left-hand side of these equations is a quadratic differential. Various qualitative characteristics of the functions realizing the extremum can be obtained by investigating the properties of the corresponding quadratic differentials. In particular, it turns out that for a large number of extremal problems in the class $ S $( and in other classes), the extremal function maps the disc $ \Delta $ onto the whole plane with a finite number of analytic slits. Sometimes the differential equation for the extremal function can be integrated, and one obtains the extremal quantity and all extremal functions in the problem considered. More often one only obtains one or a few equations for the extremal quantity. Some results obtained by variational methods are listed below.

Suppose that $ F \in \Sigma $, that $ w _ {k} $, $ k = 1 \dots n $, $ n \geq 2 $, do not belong to the image of the domain $ \Delta ^ \prime = \{ | z | > 1 \} $ under the mapping $ w = F ( z) $, and that

$$ d _ {n} ( F ) = \ \prod _ {1 \leq k < l \leq n } | w _ {k} - w _ {l} | . $$

It has been proved that

$$ d _ {3} ( F ) = \ | ( w _ {1} - w _ {2} ) ( w _ {2} - w _ {3} ) ( w _ {3} - w _ {1} ) | \leq 12 \sqrt 3 , $$

with equality only for

$$ F ( z) = z ( 1 + e ^ {i \alpha } z ^ {-} 3 ) ^ {2/3} , $$

where $ \alpha $ is real (see [1]).

It has been proved (see [1]) that the range of values of the functional

$$ w = \sum _ {\nu , \nu ^ \prime = 1 } ^ { n } \gamma _ \nu \gamma _ {\nu ^ \prime } \mathop{\rm ln} \ \frac{F ( \zeta _ \nu ) - F ( \zeta _ {\nu ^ \prime } ) }{\zeta _ \nu - \zeta _ {\nu ^ \prime } } $$

for $ F \in \Sigma $, where $ \gamma _ \nu $ are given numbers not all zero and $ \zeta _ \nu $ are given points in $ \Delta ^ \prime $, is the disc

$$ | w | \leq - \mathop{\rm Re} \ \sum _ {\nu , \nu ^ \prime = 1 } ^ { n } \gamma _ \nu \overline \gamma \; _ {\nu ^ \prime } \mathop{\rm ln} ( 1 - \zeta _ \nu ^ {-} 1 {\zeta _ {\nu ^ \prime } ^ {-} 1 } bar ) . $$

The problem has been investigated of extremizing $ | c _ {n} | $, $ n \geq 1 $, in the class $ S _ {a} $ of functions $ f ( z) = c _ {1} z + c _ {2} z ^ {2} + \dots $ that are regular and univalent in the disc $ \Delta $ and do not take on given values $ a _ {1} \dots a _ {n} $ in $ \Delta $( see [1]). The special case $ n = 1 $ is the problem of determining a continuum of least capacity (for a consideration of this problem and its generalizations see [7]).

Various problems for non-overlapping domains have been investigated by a variational method. Thus, the problem of maximizing the product

$$ I _ {n} = \ \prod _ { k= } 1 ^ { n } | f _ {k} ^ { \prime } ( 0) | $$

in the class $ \mathfrak M ( a _ {1} \dots a _ {n} ) $ has been considered (see [1]). A precise bound has been obtained for the product

$$ \prod _ { k= } 1 ^ { n } | f _ {k} ^ { \prime } ( 0) | ^ {\alpha _ {k} } , $$

where $ \alpha _ {k} $ are any given positive numbers, for $ n = 2 $ and 3 (see [1]). This problem is equivalent to the problem of finding the range $ D $ of the system of functionals $ ( | f _ {1} ( 0) | \dots | f _ {n} ( 0) | ) $ in the class $ \mathfrak M ( a _ {1} \dots a _ {n} ) $.

See also Variational principles (in complex function theory); Variation of a univalent function; Internal variations, method of; Boundary variation, method of; Variation-parametric method.

6. The method of the extremal metric.

In the solution of extremal problems by the method of the extremal metric, a fundamental role is played, as a rule, by the metric generated by a certain quadratic differential $ Q ( z) d z ^ {2} $. This is the same quadratic differential that arises in the solution of the problem by the variational method. As an example, two results obtained by this method are given below (see [1], [7][9]).

By means of the general coefficient theorem, J.A. Jenkins (1960) has solved the problem of the range of values of the functional $ f ( z) $ for fixed $ z $ in the disc $ \Delta = \{ | z | < 1 \} $ in the class $ S _ {r} $ of functions in $ S $ with real coefficients $ c _ {2} , c _ {3} ,\dots $. In the classes $ \Sigma $ and $ M $, where $ M $ is the class of functions $ f $, $ f ( 0) = 0 $, $ f ^ { \prime } ( 0) = 1 $, that are meromorphic and univalent in the disc $ \Delta $, he clarified the influence of the vanishing of a certain number of the initial coefficients on the growth of the subsequent ones.

A supplement to the general coefficient theorem has been given in the case when the differential $ Q ( z) d z ^ {2} $ has no poles of order higher than one; in addition, by means of the extremal-metric approach, very general theorems have been established on the covering of curves under a univalent conformal mapping of simply- and doubly-connected domains, including, in particular, a refinement of the result on covering of intervals for functions meromorphic and univalent on the disc, and an analogous result for a circular annulus (see [1]).

See also Grötzsch principle; Grötzsch theorems; Strip method (analytic functions); Quadratic differential; Bieberbach–Eilenberg functions; Extremal metric, method of the.

7. The method of symmetrization.

Several complicated extremal problems not lending themselves to solution by other methods have been solved by this method, often in conjunction with others. For example, the following problems are of this kind (see [1], [7][10]). For functions $ f $ in the class $ S $, a sharp upper bound has been found for the set of points of the circle $ | w | = R $, $ 1 / 4 \leq R < 1 $, not belonging to the image of the disc $ \Delta $ under the mapping $ w = f ( z) $. In conjunction with the method of the extremal metric, a sharp upper bound has been found for $ | f ( z) | $ for fixed $ | z | = r $, $ 0 < r < 1 $, for

$$ f ( z) = z + c _ {2} z ^ {2} + \dots \in S $$

with given $ c _ {2} = c = \textrm{ const } $, $ 0 \leq c \leq 2 $; the inequalities (6) have been generalized and extended to the class of functions that are $ p $- valent in the mean on the circle:

$$ f ( z) = z ^ {p} + c _ {p+} 1 z ^ {p+} 1 + \dots $$

(see Multivalent function).

By the method of symmetrization it has been proved that if $ \phi $ is a convex and non-decreasing function on $ ( - \infty , + \infty ) $, then for $ f \in S $ and $ 0 < r < 1 $,

$$ \int\limits _ {- \pi } ^ \pi \Phi ( \mathop{\rm ln} | f ( r e ^ {it} ) | ) d t \leq \int\limits _ {- \pi } ^ \pi \Phi ( \mathop{\rm ln} | K ( r e ^ {it} ) | ) d t , $$

where $ K ( z) = z ( 1 - z ) ^ {-} 2 $( see [11]). If equality holds for some $ r $, $ 0 < r < 1 $, and for some strictly convex function $ \phi $, then

$$ f ( z) = e ^ {i \alpha } K ( e ^ {i \alpha } z ) , $$

where $ \alpha $ is real.

See [12], [13] for applications of the method of symmetrization to multiply-connected domains. See also Symmetrization method.

References

[1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2] N.A. Lebedev, I.A. Aleksandrov, "On the variational method in classes of functions representable by means of Stieltjes integrals" Proc. Steklov Inst. Math. , 94 (1969) pp. 91–104 Trudy Mat. Inst. Steklov. , 94 (1968) pp. 79–89
[3] L. Brickman, T.H. MacGregor, D.R. Wilken, "Convex hulls of some classical families of univalent functions" Trans. Amer. Math. Soc. , 156 (1971) pp. 91–107
[4] N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian)
[5] I.M. Milin, "Univalent functions and orthonormal systems" , Transl. Math. Monogr. , 49 , Amer. Math. Soc. (1977) (Translated from Russian)
[6] I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)
[7] G.V. Kuz'mina, "Moduli of families of curves and quadratic differentials" Proc. Steklov Inst. Math. , 139 (1982) Trudy Mat. Inst. Steklov. , 139 (1980) pp. 1–241
[8] J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)
[9] C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[10] W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)
[11] A. Baernstein, "Integral means, univalent functions and circular symmetrization" Acta Math. , 133 (1974) pp. 139–169
[12] O.P. Mityuk, "The symmetrization principle for multiply-connected domains and certain of its applications" Ukrain. Mat. Zh. , 17 : 4 (1965) pp. 46–54 (In Russian)
[13] I.P. Mityuk, "The symmetrization principle for an annulus and certain of its applications" Sibirsk. Mat. Zh. , 6 : 6 (1965) pp. 1282–1291 (In Russian)

Comments

Instead of "univalent function" the phrase "Schlicht function" is sometimes used.

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a2] A.W. Goodman, "Univalent functions" , 1–2 , Mariner (1983)
How to Cite This Entry:
Univalent function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Univalent_function&oldid=49087
This article was adapted from an original article by N.A. Lebedev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article