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''conditions for univalence''
 
''conditions for univalence''
  
Necessary and sufficient conditions for a regular (or meromorphic) function to be univalent in a domain of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956001.png" /> (cf. [[Univalent function|Univalent function]]). A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956002.png" /> to be univalent in a sufficiently small neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956003.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956004.png" />. Such (local) univalence at every point of a domain does not yet ensure univalence in the domain. For example, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956005.png" /> is not univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956007.png" />, although it satisfies the condition for local univalence at every point of the plane. Any property of univalent functions, and in particular any inequality satisfied by all univalent functions, is a necessary condition for univalence. The following are necessary and sufficient conditions for univalence.
+
Necessary and sufficient conditions for a regular (or meromorphic) function to be univalent in a domain of the complex plane $  \mathbf C $ (cf. [[Univalent function|Univalent function]]). A necessary and sufficient condition for $  f ( z) $
 +
to be univalent in a sufficiently small neighbourhood of a point $  a $
 +
is that $  f ^ { \prime } ( a) \neq 0 $.  
 +
Such (local) univalence at every point of a domain does not yet ensure univalence in the domain. For example, the function $  e  ^ {z} $
 +
is not univalent in the disc $  | z | \leq  R $,  
 +
where $  R > \pi $,  
 +
although it satisfies the condition for local univalence at every point of the plane. Any property of univalent functions, and in particular any inequality satisfied by all univalent functions, is a necessary condition for univalence. The following are necessary and sufficient conditions for univalence.
  
 
===Theorem 1.===
 
===Theorem 1.===
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956008.png" /> has a series expansion
+
Suppose that $  f ( z) $
 +
has a series expansion
 +
 
 +
$$ \tag{1 }
 +
f ( z)  =  z + a _ {2} z  ^ {2} + \dots + a _ {n} z  ^ {n} + \dots
 +
$$
  
<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/u095600/u0956009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
in a neighbourhood of  $  z = 0 $,
 +
and let
  
in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560010.png" />, and let
+
$$
 +
\mathop{\rm ln} \
  
<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/u095600/u09560011.png" /></td> </tr></table>
+
\frac{f ( t) - f ( z) }{t - z }
 +
  = \
 +
\sum _ {p , q = 0 } ^  \infty 
 +
\omega _ {p,q} t  ^ {p} z  ^ {q}
 +
$$
  
with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560013.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560014.png" /> to be regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560015.png" /> it is necessary and sufficient that for every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560016.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560018.png" />, the Grunsky inequalities are satisfied:
+
with constant coefficients $  a _ {k} $
 +
and $  \omega _ {p,q} $.  
 +
For $  f ( z) $
 +
to be regular and univalent in $  E = \{ {z } : {| z | < 1 } \} $
 +
it is necessary and sufficient that for every positive integer $  N $
 +
and all $  x _ {p} $,  
 +
$  p = 1 \dots N $,  
 +
the Grunsky inequalities are satisfied:
  
<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/u095600/u09560019.png" /></td> </tr></table>
+
$$
 +
\left | \sum _ {p , q = 1 } ^ { N }
 +
\omega _ {p,q} x _ {p} x _ {q} \right |  \leq  \
 +
\sum _ { p= 1} ^ { N } 
 +
\frac{1}{p}
 +
| x _ {p} |  ^ {2} .
 +
$$
  
Similar conditions hold for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560021.png" /> (the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560022.png" /> that are meromorphic and univalent in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560023.png" />; see [[#References|[2]]], and also [[Area principle|Area principle]]).
+
Similar conditions hold for the class $  \Sigma ( B) $ (the class of functions $  F ( \zeta ) = \zeta + c _ {0} + c _ {1} / \zeta + \dots $
 +
that are meromorphic and univalent in a domain $  B \ni \infty $;  
 +
see [[#References|[2]]], and also [[Area principle|Area principle]]).
  
 
===Theorem 2.===
 
===Theorem 2.===
Let the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560024.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560025.png" /> be a Jordan curve. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560026.png" /> be regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560027.png" /> and continuous on the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560028.png" />. A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560029.png" /> to be univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560030.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560031.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560032.png" /> bijectively onto some closed Jordan curve.
+
Let the boundary $  l $
 +
of a bounded domain $  D $
 +
be a Jordan curve. Let the function $  f ( z) $
 +
be regular in $  D $
 +
and continuous on the closed domain $  \overline{D} $.  
 +
A necessary and sufficient condition for $  f ( z) $
 +
to be univalent in $  \overline{D} $
 +
is that $  f $
 +
maps $  l $
 +
bijectively onto some closed Jordan curve.
  
Necessary and sufficient conditions for the function (1) on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560033.png" /> to be a univalent mapping onto a convex domain, or a domain star-like or spiral-like relative to the origin, are related to theorem 2, and can be stated, respectively, in the forms
+
Necessary and sufficient conditions for the function (1) on the disc $  E $
 +
to be a univalent mapping onto a convex domain, or a domain star-like or spiral-like relative to the origin, are related to theorem 2, and can be stated, respectively, in the forms
  
<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/u095600/u09560034.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re}
 +
\left ( z
 +
\frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) }
 +
\right ) + 1  \geq  0 ,\  \mathop{\rm Re}
 +
\left ( z
 +
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
\right )  \geq  0 ,
 +
$$
  
<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/u095600/u09560035.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re} \left ( e ^ {i \gamma } z
 +
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
\right )  \geq  0 .
 +
$$
  
 
Many sufficient univalence conditions can be described by means of ordinary (theorem 3) or partial (theorem 4) differential equations.
 
Many sufficient univalence conditions can be described by means of ordinary (theorem 3) or partial (theorem 4) differential equations.
  
 
===Theorem 3.===
 
===Theorem 3.===
A meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560036.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560037.png" /> is univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560038.png" /> if the Schwarzian derivative
+
A meromorphic function $  f ( z) $
 +
in the disc $  E $
 +
is univalent in $  E $
 +
if the Schwarzian derivative
  
<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/u095600/u09560039.png" /></td> </tr></table>
+
$$
 +
\{ f , z \}  = \
 +
\left [
 +
\frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) }
 +
\right ]  ^  \prime
 +
-  
 +
\frac{1}{2}
 +
\left [
 +
\frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) }
 +
\right ]  ^ {2}
 +
$$
  
 
satisfies the inequality
 
satisfies 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/u095600/u09560040.png" /></td> </tr></table>
+
$$
 +
| \{ f , z \} |  \leq  2 S ( | z | ) ,\ \
 +
| z | < 1 ,
 +
$$
  
where the majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560041.png" /> is a non-negative continuous function satisfying the conditions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560042.png" /> does not increase in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560043.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560044.png" />; and b) the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560046.png" /> has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560047.png" />.
+
where the majorant $  S ( r) $
 +
is a non-negative continuous function satisfying the conditions: a) $  S ( r) ( 1 - r  ^ {2} )  ^ {2} $
 +
does not increase in $  r $
 +
for  $  0 < r < 1 $;  
 +
and b) the differential equation $  y  ^ {\prime\prime} + S ( | t | ) y = 0 $
 +
for $  - 1 < t < 1 $
 +
has a solution $  y _ {0} ( t) > 0 $.
  
 
A special case of theorem 3 is formed by the Nehari–Pokornii univalence conditions:
 
A special case of theorem 3 is formed by the Nehari–Pokornii univalence conditions:
  
<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/u095600/u09560048.png" /></td> </tr></table>
+
$$
 +
| \{ f , z \} |  \leq 
 +
\frac{C ( \mu ) }{( 1 - | z |  ^ {2} )  ^  \mu  }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560049.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560051.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560052.png" />.
+
where $  C ( \mu ) = 2 ^ {3 \mu - 1 } \pi ^ {2 ( 1 - \mu ) } $
 +
if 0 \leq  \mu \leq  1 $
 +
and = 2 ^ {3 - \mu } $
 +
if $  1 \leq  \mu \leq  2 $.
  
 
===Theorem 4.===
 
===Theorem 4.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560053.png" /> be a regular function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560054.png" /> that is continuously differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560057.png" />, and satisfying the Löwner–Kufarev equation
+
Let $  f ( z , t ) $
 +
be a regular function in the disc $  E $
 +
that is continuously differentiable with respect to $  t $,
 +
$  0 \leq  t < \infty $,  
 +
$  f ( 0 , t ) = 0 $,  
 +
and satisfying the Löwner–Kufarev equation
 +
 
 +
$$
 +
 
 +
\frac{\partial  f }{\partial  t }
 +
 
 +
=  z h ( z , t )
  
<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/u095600/u09560058.png" /></td> </tr></table>
+
\frac{\partial  f }{\partial  z }
 +
,\ \
 +
0 < t < \infty ,\ \
 +
z \in E ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560059.png" /> is a regular function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560060.png" />, continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560063.png" />. If
+
where $  h ( z , t ) $
 +
is a regular function in $  E $,  
 +
continuous in $  t $,
 +
0 \leq  t < \infty $,  
 +
and $  \mathop{\rm Re}  h ( z , t ) \geq  0 $.  
 +
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/u095600/u09560064.png" /></td> </tr></table>
+
$$
 +
f ( z , t )  = a _ {0} ( t) f ( z) + O ( 1) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560066.png" /> is a bounded quantity as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560067.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560068.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560069.png" /> is a regular non-constant function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560070.png" /> with expansion (1), then all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560071.png" /> are univalent, including the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560073.png" />.
+
where $  \lim\limits _ {t \rightarrow \infty }  a _ {0} ( t) = \infty $,  
 +
$  O ( 1) $
 +
is a bounded quantity as $  t \rightarrow \infty $
 +
for every $  z \in E $,  
 +
and $  f ( z) $
 +
is a regular non-constant function on $  E $
 +
with expansion (1), then all functions $  f ( z , t ) $
 +
are univalent, including the functions $  f ( z , 0 ) $
 +
and $  f ( z) $.
  
 
Theorem 4 implies the following special univalence conditions:
 
Theorem 4 implies the following special univalence conditions:
  
<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/u095600/u09560074.png" /></td> </tr></table>
+
$$
 +
\left | z
 +
\frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) }
 +
\right |  \leq  \
 +
 
 +
\frac{1}{1 - | z |  ^ {2} }
 +
 
 +
$$
  
 
and
 
and
  
<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/u095600/u09560075.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re} \left [
 +
e ^ {i \gamma } \left (
 +
\frac{f ( z) }{z}
 +
\right ) ^ {\alpha + i \beta - 1 }
 +
 
 +
\frac{f ^ { \prime } ( z) }{\phi ^ {\prime \alpha } ( z) }
 +
 
 +
\right ]  \geq  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560078.png" /> are real constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560080.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560081.png" /> is a regular function mapping the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560082.png" /> onto a convex domain.
+
where $  \alpha $,  
 +
$  \beta $,  
 +
$  \gamma $
 +
are real constants, $  \alpha > 0 $,  
 +
$  | \gamma | < \pi / 2 $,  
 +
and $  \phi ( z) $
 +
is a regular function mapping the disc $  E $
 +
onto a convex domain.
  
 
The univalence of the function
 
The univalence of the 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/u095600/u09560083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= f ( z)
 +
$$
  
is equivalent to the uniqueness of the solution of (2) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560084.png" />. In this sense, sufficient univalence conditions can be extended to a wide class of operator equations. For these equations, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560085.png" /> can, in particular, be generalized to a class of real mappings of domains in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560086.png" />-dimensional Euclidean space.
+
is equivalent to the uniqueness of the solution of (2) in $  z $.  
 +
In this sense, sufficient univalence conditions can be extended to a wide class of operator equations. For these equations, the condition $  \mathop{\rm Re} [ e ^ {i \gamma } f ^ { \prime } ( z) ] \geq  0 $
 +
can, in particular, be generalized to a class of real mappings of domains in an $  n $-dimensional Euclidean space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</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">[3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.G. Avkhadiev,  L.A. Aksent'ev,  "The main results on sufficient conditions for an analytic function to be schlicht"  ''Russian Math. Surveys'' , '''30''' :  4  (1975)  pp. 1–64  ''Uspekhi Mat. Nauk'' , '''30''' :  4  (1975)  pp. 3–60</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F.D. Gakhov,  "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.G. Tumashev,  M.T. Nuzhin,  "Inverse boundary value problems and their applications" , Kazan'  (1965)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</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">[3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.G. Avkhadiev,  L.A. Aksent'ev,  "The main results on sufficient conditions for an analytic function to be schlicht"  ''Russian Math. Surveys'' , '''30''' :  4  (1975)  pp. 1–64  ''Uspekhi Mat. Nauk'' , '''30''' :  4  (1975)  pp. 3–60</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F.D. Gakhov,  "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.G. Tumashev,  M.T. Nuzhin,  "Inverse boundary value problems and their applications" , Kazan'  (1965)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:59, 10 May 2022


conditions for univalence

Necessary and sufficient conditions for a regular (or meromorphic) function to be univalent in a domain of the complex plane $ \mathbf C $ (cf. Univalent function). A necessary and sufficient condition for $ f ( z) $ to be univalent in a sufficiently small neighbourhood of a point $ a $ is that $ f ^ { \prime } ( a) \neq 0 $. Such (local) univalence at every point of a domain does not yet ensure univalence in the domain. For example, the function $ e ^ {z} $ is not univalent in the disc $ | z | \leq R $, where $ R > \pi $, although it satisfies the condition for local univalence at every point of the plane. Any property of univalent functions, and in particular any inequality satisfied by all univalent functions, is a necessary condition for univalence. The following are necessary and sufficient conditions for univalence.

Theorem 1.

Suppose that $ f ( z) $ has a series expansion

$$ \tag{1 } f ( z) = z + a _ {2} z ^ {2} + \dots + a _ {n} z ^ {n} + \dots $$

in a neighbourhood of $ z = 0 $, and let

$$ \mathop{\rm ln} \ \frac{f ( t) - f ( z) }{t - z } = \ \sum _ {p , q = 0 } ^ \infty \omega _ {p,q} t ^ {p} z ^ {q} $$

with constant coefficients $ a _ {k} $ and $ \omega _ {p,q} $. For $ f ( z) $ to be regular and univalent in $ E = \{ {z } : {| z | < 1 } \} $ it is necessary and sufficient that for every positive integer $ N $ and all $ x _ {p} $, $ p = 1 \dots N $, the Grunsky inequalities are satisfied:

$$ \left | \sum _ {p , q = 1 } ^ { N } \omega _ {p,q} x _ {p} x _ {q} \right | \leq \ \sum _ { p= 1} ^ { N } \frac{1}{p} | x _ {p} | ^ {2} . $$

Similar conditions hold for the class $ \Sigma ( B) $ (the class of functions $ F ( \zeta ) = \zeta + c _ {0} + c _ {1} / \zeta + \dots $ that are meromorphic and univalent in a domain $ B \ni \infty $; see [2], and also Area principle).

Theorem 2.

Let the boundary $ l $ of a bounded domain $ D $ be a Jordan curve. Let the function $ f ( z) $ be regular in $ D $ and continuous on the closed domain $ \overline{D} $. A necessary and sufficient condition for $ f ( z) $ to be univalent in $ \overline{D} $ is that $ f $ maps $ l $ bijectively onto some closed Jordan curve.

Necessary and sufficient conditions for the function (1) on the disc $ E $ to be a univalent mapping onto a convex domain, or a domain star-like or spiral-like relative to the origin, are related to theorem 2, and can be stated, respectively, in the forms

$$ \mathop{\rm Re} \left ( z \frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) } \right ) + 1 \geq 0 ,\ \mathop{\rm Re} \left ( z \frac{f ^ { \prime } ( z) }{f ( z) } \right ) \geq 0 , $$

$$ \mathop{\rm Re} \left ( e ^ {i \gamma } z \frac{f ^ { \prime } ( z) }{f ( z) } \right ) \geq 0 . $$

Many sufficient univalence conditions can be described by means of ordinary (theorem 3) or partial (theorem 4) differential equations.

Theorem 3.

A meromorphic function $ f ( z) $ in the disc $ E $ is univalent in $ E $ if the Schwarzian derivative

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

satisfies the inequality

$$ | \{ f , z \} | \leq 2 S ( | z | ) ,\ \ | z | < 1 , $$

where the majorant $ S ( r) $ is a non-negative continuous function satisfying the conditions: a) $ S ( r) ( 1 - r ^ {2} ) ^ {2} $ does not increase in $ r $ for $ 0 < r < 1 $; and b) the differential equation $ y ^ {\prime\prime} + S ( | t | ) y = 0 $ for $ - 1 < t < 1 $ has a solution $ y _ {0} ( t) > 0 $.

A special case of theorem 3 is formed by the Nehari–Pokornii univalence conditions:

$$ | \{ f , z \} | \leq \frac{C ( \mu ) }{( 1 - | z | ^ {2} ) ^ \mu } , $$

where $ C ( \mu ) = 2 ^ {3 \mu - 1 } \pi ^ {2 ( 1 - \mu ) } $ if $ 0 \leq \mu \leq 1 $ and $ = 2 ^ {3 - \mu } $ if $ 1 \leq \mu \leq 2 $.

Theorem 4.

Let $ f ( z , t ) $ be a regular function in the disc $ E $ that is continuously differentiable with respect to $ t $, $ 0 \leq t < \infty $, $ f ( 0 , t ) = 0 $, and satisfying the Löwner–Kufarev equation

$$ \frac{\partial f }{\partial t } = z h ( z , t ) \frac{\partial f }{\partial z } ,\ \ 0 < t < \infty ,\ \ z \in E , $$

where $ h ( z , t ) $ is a regular function in $ E $, continuous in $ t $, $ 0 \leq t < \infty $, and $ \mathop{\rm Re} h ( z , t ) \geq 0 $. If

$$ f ( z , t ) = a _ {0} ( t) f ( z) + O ( 1) , $$

where $ \lim\limits _ {t \rightarrow \infty } a _ {0} ( t) = \infty $, $ O ( 1) $ is a bounded quantity as $ t \rightarrow \infty $ for every $ z \in E $, and $ f ( z) $ is a regular non-constant function on $ E $ with expansion (1), then all functions $ f ( z , t ) $ are univalent, including the functions $ f ( z , 0 ) $ and $ f ( z) $.

Theorem 4 implies the following special univalence conditions:

$$ \left | z \frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) } \right | \leq \ \frac{1}{1 - | z | ^ {2} } $$

and

$$ \mathop{\rm Re} \left [ e ^ {i \gamma } \left ( \frac{f ( z) }{z} \right ) ^ {\alpha + i \beta - 1 } \frac{f ^ { \prime } ( z) }{\phi ^ {\prime \alpha } ( z) } \right ] \geq 0 , $$

where $ \alpha $, $ \beta $, $ \gamma $ are real constants, $ \alpha > 0 $, $ | \gamma | < \pi / 2 $, and $ \phi ( z) $ is a regular function mapping the disc $ E $ onto a convex domain.

The univalence of the function

$$ \tag{2 } w = f ( z) $$

is equivalent to the uniqueness of the solution of (2) in $ z $. In this sense, sufficient univalence conditions can be extended to a wide class of operator equations. For these equations, the condition $ \mathop{\rm Re} [ e ^ {i \gamma } f ^ { \prime } ( z) ] \geq 0 $ can, in particular, be generalized to a class of real mappings of domains in an $ n $-dimensional Euclidean space.

References

[1] N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian)
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
[4] F.G. Avkhadiev, L.A. Aksent'ev, "The main results on sufficient conditions for an analytic function to be schlicht" Russian Math. Surveys , 30 : 4 (1975) pp. 1–64 Uspekhi Mat. Nauk , 30 : 4 (1975) pp. 3–60
[5] F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)
[6] G.G. Tumashev, M.T. Nuzhin, "Inverse boundary value problems and their applications" , Kazan' (1965) (In Russian)

Comments

Instead of "univalence" the German word "Schlicht" is sometimes used, also in the English language literature.

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
How to Cite This Entry:
Univalency conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Univalency_conditions&oldid=15592
This article was adapted from an original article by L.A. Aksent'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article