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''for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229202.png" />''
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''for the class  $  S $''
  
 
A problem for the class of functions
 
A problem for the class of 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/c/c022/c022920/c0229203.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \
 +
z + \sum _ {n = 2 } ^  \infty 
 +
c _ {n} z  ^ {n}
 +
$$
 +
 
 +
which are regular and univalent in the disc  $  | z | < 1 $.  
 +
It consists of determining for every  $  n $,
 +
$  n \geq  2 $,
 +
the region of values  $  V _ {n} $
 +
for the system of coefficients  $  \{ c _ {2} \dots c _ {n} \} $
 +
of the functions of this class and, in particular, to find sharp bounds for  $  | c _ {n} | $,
 +
$  n \geq  2 $,
 +
in the class  $  S $(
 +
see [[Bieberbach conjecture|Bieberbach conjecture]]). The coefficient problem for a class  $  R $
 +
of functions regular in  $  | z | < 1 $
 +
consists in determining in  $  R $,
 +
for every  $  n $,
 +
$  n \geq  1 $,
 +
the region of values of the first  $  n $
 +
coefficients in the series expansions of the functions of  $  R $
 +
in powers of  $  z $
 +
and, in particular, in obtaining sharp bounds for these coefficients in the class  $  R $.
 +
The coefficient problem has been solved for the [[Carathéodory class|Carathéodory class]], for the class of univalent star-like functions, and for the class of functions regular and bounded in  $  | z | < 1 $.
  
which are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229204.png" />. It consists of determining for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229206.png" />, the region of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229207.png" /> for the system of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229208.png" /> of the functions of this class and, in particular, to find sharp bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c0229209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292010.png" />, in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292011.png" /> (see [[Bieberbach conjecture|Bieberbach conjecture]]). The coefficient problem for a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292012.png" /> of functions regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292013.png" /> consists in determining in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292014.png" />, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292016.png" />, the region of values of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292017.png" /> coefficients in the series expansions of the functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292018.png" /> in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292019.png" /> and, in particular, in obtaining sharp bounds for these coefficients in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292020.png" />. The coefficient problem has been solved for the [[Carathéodory class|Carathéodory class]], for the class of univalent star-like functions, and for the class of functions regular and bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292021.png" />.
+
It is known that  $  V _ {2} $
 +
is a disc: $  | c _ {2} | \leq  2 $.  
 +
Profound qualitative results with regard to the coefficient problem have been obtained for the class  $  S $(
 +
see [[#References|[7]]]). The set  $  V _ {n} $
 +
is a bounded closed domain; the point  $  c _ {2} = 0 \dots c _ {n} = 0 $
 +
is an interior point of  $  V _ {n} $;
 +
$  V _ {n} $
 +
is homeomorphic to a closed  $  ( 2n - 2) $-
 +
dimensional ball; the boundary of $  V _ {n} $
 +
is a union of finitely many parts  $  \Pi _ {1} \dots \Pi _ {N} $;
 +
the coordinates of a point  $  ( c _ {2} \dots c _ {n} ) $
 +
on any one of these parts are functions of a finite number ( $  \leq  2n - 3 $)
 +
of parameters. To every boundary point of  $  V _ {n} $
 +
there corresponds a unique function of the class $  S $.  
 +
The boundary of  $  V _ {3} $
 +
is a union of two hyperplanes  $  \Pi _ {1} $
 +
and  $  \Pi _ {2} $
 +
of dimension 3 and their intersections: planes  $  \Pi _ {3} $
 +
and  $  \Pi _ {4} $
 +
and a curve  $  \Pi _ {5} $.  
 +
Parametric formulas have been derived for  $  \Pi _ {1} $
 +
and  $  \Pi _ {2} $
 +
in terms of elementary functions. The intersection of  $  V _ {3} $
 +
with the plane  $  \mathop{\rm Im}  c _ {2} = 0 $
 +
is symmetric about the planes  $  \mathop{\rm Re}  c _ {2} = 0 $
 +
and  $  \mathop{\rm Im}  c _ {3} = 0 $.  
 +
The intersection of $  V _ {3} $
 +
with the plane  $  \mathop{\rm Im}  c _ {3} = 0 $
 +
is symmetric about the planes  $  \mathop{\rm Im}  c _ {2} = 0 $
 +
and  $  \mathop{\rm Re}  c _ {2} = 0 $.
 +
A function  $  w = f ( z) $
 +
corresponding to a point on  $  \Pi _ {1} $
 +
maps  $  | z | < 1 $
 +
onto the  $  w $-
 +
plane cut by an analytic curve going to infinity. A function  $  w = f ( z) $
 +
corresponding to a point on  $  \Pi _ {2} $
 +
maps  $  | z | < 1 $
 +
onto the $  w $-
 +
plane cut by three analytic arcs, issuing from a finite point and inclined to one another at angles  $  2 \pi /3 $;
 +
one of these arcs lies on a straight line  $  \mathop{\rm arg}  w = \textrm{ const } $
 +
and goes to infinity.
  
It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292022.png" /> is a disc: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292023.png" />. Profound qualitative results with regard to the coefficient problem have been obtained for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292024.png" /> (see [[#References|[7]]]). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292025.png" /> is a bounded closed domain; the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292026.png" /> is an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292027.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292028.png" /> is homeomorphic to a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292029.png" />-dimensional ball; the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292030.png" /> is a union of finitely many parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292031.png" />; the coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292032.png" /> on any one of these parts are functions of a finite number (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292033.png" />) of parameters. To every boundary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292034.png" /> there corresponds a unique function of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292035.png" />. The boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292036.png" /> is a union of two hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292038.png" /> of dimension 3 and their intersections: planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292040.png" /> and a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292041.png" />. Parametric formulas have been derived for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292043.png" /> in terms of elementary functions. The intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292044.png" /> with the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292045.png" /> is symmetric about the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292047.png" />. The intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292048.png" /> with the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292049.png" /> is symmetric about the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292051.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292052.png" /> corresponding to a point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292053.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292054.png" /> onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292055.png" />-plane cut by an analytic curve going to infinity. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292056.png" /> corresponding to a point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292057.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292058.png" /> onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292059.png" />-plane cut by three analytic arcs, issuing from a finite point and inclined to one another at angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292060.png" />; one of these arcs lies on a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292061.png" /> and goes to infinity.
+
Among the other special regions that have been investigated are the following: the region of values  $  \{ c _ {2} , c _ {3} \} $
 +
in the subclass of $  S $
 +
consisting of functions with real  $  c _ {2} $
 +
and c _ {3} $;
 +
the region of values  $  \{ | c _ {k + 1 }  |, | c _ {2k + 1 }  | \} $
 +
and $  \{ c _ {k + 1 }  , c _ {2k + 1 }  \} $,
 +
if  $  \mathop{\rm Im}  c _ {k + 1 }  = \mathop{\rm Im}  c _ {2k + 1 }  = 0 $,
 +
on the subclass of bounded functions in  $  S $
 +
representable as
  
Among the other special regions that have been investigated are the following: the region of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292062.png" /> in the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292063.png" /> consisting of functions with real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292065.png" />; the region of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292067.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292068.png" />, on the subclass of bounded functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292069.png" /> representable as
+
$$
 +
f ( z)  = z +
 +
\sum _ {n = 1 } ^  \infty 
 +
c _ {nk + 1 }
 +
z ^ {nk + 1 } ;
 +
$$
  
<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/c/c022/c022920/c02292070.png" /></td> </tr></table>
+
the region of values  $  \{ c _ {2} , c _ {3} \} $
 +
on the subclass of bounded functions in  $  S $;  
 +
the region of values  $  \{ c _ {2} , c _ {3} , c _ {4} \} $
 +
on the subclass of functions in  $  S $
 +
with real  $  c _ {2} , c _ {3} $
 +
and  $  c _ {4} $.
  
the region of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292071.png" /> on the subclass of bounded functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292072.png" />; the region of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292073.png" /> on the subclass of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292074.png" /> with real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292076.png" />.
+
Sharp bounds for the coefficients, of the type  $  | c _ {n} | \leq  A _ {n} $,
 +
$  n \geq  2 $,
 +
have been obtained in the subclass of convex functions in $  S $
 +
with  $  A _ {n} = 1 $(
 +
cf. [[Convex function (of a complex variable)|Convex function (of a complex variable)]]), in the subclass of star-like functions in  $  S $
 +
with  $  A _ {n} = n $,
 +
in the subclass of odd star-like functions in $  S $
 +
with  $  A _ {n} = 1 $,
 +
$  n = 3, 5 \dots $
 +
in the class of univalent functions having real coefficients with  $  A _ {n} = n $,
 +
in the subclass of close-to-convex functions in  $  S $
 +
with $  A _ {n} = n $,
 +
and in the class  $  S $
 +
itself with  $  A _ {n} = n $(
 +
cf. [[Bieberbach conjecture|Bieberbach conjecture]], [[#References|[8]]]). In the class of functions
  
Sharp bounds for the coefficients, of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292078.png" />, have been obtained in the subclass of convex functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292079.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292080.png" /> (cf. [[Convex function (of a complex variable)|Convex function (of a complex variable)]]), in the subclass of star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292081.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292082.png" />, in the subclass of odd star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292083.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292085.png" /> in the class of univalent functions having real coefficients with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292086.png" />, in the subclass of close-to-convex functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292088.png" />, and in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292089.png" /> itself with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292090.png" /> (cf. [[Bieberbach conjecture|Bieberbach conjecture]], [[#References|[8]]]). In the class of functions
+
$$
 +
f ( z) = z +
 +
\sum _ {n = 2 } ^  \infty 
 +
c _ {n} z  ^ {n}
 +
$$
  
<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/c/c022/c022920/c02292091.png" /></td> </tr></table>
+
which are regular and typically real in  $  | z | < 1 $
 +
one has the sharp bound  $  | c _ {n} | \leq  n $,
 +
$  n \geq  2 $,
 +
and in the class of [[Bieberbach–Eilenberg functions|Bieberbach–Eilenberg functions]]  $  f ( z) = a _ {1} z + a _ {2} z  ^ {2} + \dots $
 +
one has the sharp bound  $  | a _ {n} | \leq  1 $,
 +
$  n \geq  1 $.
  
which are regular and typically real in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292092.png" /> one has the sharp bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292094.png" />, and in the class of [[Bieberbach–Eilenberg functions|Bieberbach–Eilenberg functions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292095.png" /> one has the sharp bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292097.png" />.
+
Sharp bounds are known for the class $  \Sigma $
 +
of functions
  
Sharp bounds are known for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c02292098.png" /> of functions
+
$$
 +
F ( \zeta )  = \
 +
\zeta +
 +
\sum _ {n = 0 } ^  \infty 
  
<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/c/c022/c022920/c02292099.png" /></td> </tr></table>
+
\frac{b _ {n} }{\zeta  ^ {n} }
  
which are meromorphic and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920100.png" />; these are
+
$$
  
<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/c/c022/c022920/c022920101.png" /></td> </tr></table>
+
which are meromorphic and univalent in  $  | \zeta | > 1 $;  
 +
these are
  
For the subclass of star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920102.png" />, one has the sharp bound
+
$$
 +
| b _ {1} |  \leq  1,\ \
 +
| b _ {2} |  \leq  {
 +
\frac{2}{3}
 +
} ,\ \
 +
| b _ {3} |  \leq  {
 +
\frac{1}{2}
 +
} + e  ^ {-} 6 .
 +
$$
  
<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/c/c022/c022920/c022920103.png" /></td> </tr></table>
+
For the subclass of star-like functions in  $  \Sigma $,
 +
one has the sharp bound
  
Sharp bounds are also known for other subclasses of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920105.png" /> (see [[#References|[1]]]–[[#References|[4]]]), and also for some classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920106.png" />-valent functions and in classes of functions which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920107.png" />-valent in the mean (see [[#References|[5]]]).
+
$$
 +
| b _ {n} |  \leq  \
 +
 
 +
\frac{2}{n + 1 }
 +
,\ \
 +
n \geq  1.
 +
$$
 +
 
 +
Sharp bounds are also known for other subclasses of $  S $
 +
and $  \Sigma $(
 +
see [[#References|[1]]]–[[#References|[4]]]), and also for some classes of $  p $-
 +
valent functions and in classes of functions which are $  p $-
 +
valent in the mean (see [[#References|[5]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric methods in the 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">  I.E. Bazilevich,  , ''Mathematics in the USSR during 40 years: 1917–1957'' , '''1''' , Moscow  (1959)  pp. 444–472  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.K. Hayman,  "Coefficient problems for univalent functions and related function classes"  ''J. London Math. Soc.'' , '''40''' :  3  (1965)  pp. 385–406</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.W. Goodman,  "Open problems on univalent and multivalent functions"  ''Bull. Amer. Math. Soc.'' , '''74''' :  6  (1968)  pp. 1035–1050</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Phelps,  "On a coefficient problem in univalent functions"  ''Trans. Amer. Math. Soc.'' , '''143'''  (1969)  pp. 475–485</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.C. Schaeffer,  D.C. Spencer,  "Coefficient regions for schlicht functions" , ''Amer. Math. Soc. Coll. Publ.'' , '''35''' , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. de Branges,  "A proof of the Bieberbach conjecture"  ''Acta Math.'' , '''154''' :  1–2  (1985)  pp. 137–152</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric methods in the 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">  I.E. Bazilevich,  , ''Mathematics in the USSR during 40 years: 1917–1957'' , '''1''' , Moscow  (1959)  pp. 444–472  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.K. Hayman,  "Coefficient problems for univalent functions and related function classes"  ''J. London Math. Soc.'' , '''40''' :  3  (1965)  pp. 385–406</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.W. Goodman,  "Open problems on univalent and multivalent functions"  ''Bull. Amer. Math. Soc.'' , '''74''' :  6  (1968)  pp. 1035–1050</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Phelps,  "On a coefficient problem in univalent functions"  ''Trans. Amer. Math. Soc.'' , '''143'''  (1969)  pp. 475–485</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.C. Schaeffer,  D.C. Spencer,  "Coefficient regions for schlicht functions" , ''Amer. Math. Soc. Coll. Publ.'' , '''35''' , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. de Branges,  "A proof of the Bieberbach conjecture"  ''Acta Math.'' , '''154''' :  1–2  (1985)  pp. 137–152</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For functions in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920108.png" />, the estimates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920110.png" /> mentioned above are due to M. Schiffer [[#References|[a1]]] and P.R. Garabedian and Schiffer [[#References|[a2]]], respectively. The sharp bound for star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022920/c022920111.png" /> is due to J. Clunie [[#References|[a3]]] and C. Pommerenke [[#References|[a4]]]. Standard references in English include [[#References|[a5]]]–[[#References|[a7]]].
+
For functions in the class $  \Sigma $,  
 +
the estimates for $  | b _ {2} | $
 +
and $  | b _ {3} | $
 +
mentioned above are due to M. Schiffer [[#References|[a1]]] and P.R. Garabedian and Schiffer [[#References|[a2]]], respectively. The sharp bound for star-like functions in $  \Sigma $
 +
is due to J. Clunie [[#References|[a3]]] and C. Pommerenke [[#References|[a4]]]. Standard references in English include [[#References|[a5]]]–[[#References|[a7]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Schiffer,  "Sur un problème d'extrémum de la répresentation conforme"  ''Bull. Soc. Math. France'' , '''66'''  (1938)  pp. 48–55</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  M. Schiffer,  "A coefficient inequality for schlicht functions"  ''Ann. of Math.'' , '''61'''  (1955)  pp. 116–136</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Clunie,  "On meromorphic schlicht functions"  ''J. London Math. Soc.'' , '''34'''  (1959)  pp. 215–216</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Pommerenke,  "On meromorphic starlike functions"  ''Pacific J. Math.'' , '''13'''  (1963)  pp. 221–235</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.K. Hayman,  "Multivalent functions" , Cambridge Univ. Press  (1967)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Pommerenke,  "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. 258</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  O. Tammi,  "Extremum problems for bounded univalent functions II" , ''Lect. notes in math.'' , '''913''' , Springer  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Schiffer,  "Sur un problème d'extrémum de la répresentation conforme"  ''Bull. Soc. Math. France'' , '''66'''  (1938)  pp. 48–55</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  M. Schiffer,  "A coefficient inequality for schlicht functions"  ''Ann. of Math.'' , '''61'''  (1955)  pp. 116–136</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Clunie,  "On meromorphic schlicht functions"  ''J. London Math. Soc.'' , '''34'''  (1959)  pp. 215–216</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Pommerenke,  "On meromorphic starlike functions"  ''Pacific J. Math.'' , '''13'''  (1963)  pp. 221–235</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.K. Hayman,  "Multivalent functions" , Cambridge Univ. Press  (1967)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Pommerenke,  "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. 258</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  O. Tammi,  "Extremum problems for bounded univalent functions II" , ''Lect. notes in math.'' , '''913''' , Springer  (1982)</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


for the class $ S $

A problem for the class of functions

$$ f ( z) = \ z + \sum _ {n = 2 } ^ \infty c _ {n} z ^ {n} $$

which are regular and univalent in the disc $ | z | < 1 $. It consists of determining for every $ n $, $ n \geq 2 $, the region of values $ V _ {n} $ for the system of coefficients $ \{ c _ {2} \dots c _ {n} \} $ of the functions of this class and, in particular, to find sharp bounds for $ | c _ {n} | $, $ n \geq 2 $, in the class $ S $( see Bieberbach conjecture). The coefficient problem for a class $ R $ of functions regular in $ | z | < 1 $ consists in determining in $ R $, for every $ n $, $ n \geq 1 $, the region of values of the first $ n $ coefficients in the series expansions of the functions of $ R $ in powers of $ z $ and, in particular, in obtaining sharp bounds for these coefficients in the class $ R $. The coefficient problem has been solved for the Carathéodory class, for the class of univalent star-like functions, and for the class of functions regular and bounded in $ | z | < 1 $.

It is known that $ V _ {2} $ is a disc: $ | c _ {2} | \leq 2 $. Profound qualitative results with regard to the coefficient problem have been obtained for the class $ S $( see [7]). The set $ V _ {n} $ is a bounded closed domain; the point $ c _ {2} = 0 \dots c _ {n} = 0 $ is an interior point of $ V _ {n} $; $ V _ {n} $ is homeomorphic to a closed $ ( 2n - 2) $- dimensional ball; the boundary of $ V _ {n} $ is a union of finitely many parts $ \Pi _ {1} \dots \Pi _ {N} $; the coordinates of a point $ ( c _ {2} \dots c _ {n} ) $ on any one of these parts are functions of a finite number ( $ \leq 2n - 3 $) of parameters. To every boundary point of $ V _ {n} $ there corresponds a unique function of the class $ S $. The boundary of $ V _ {3} $ is a union of two hyperplanes $ \Pi _ {1} $ and $ \Pi _ {2} $ of dimension 3 and their intersections: planes $ \Pi _ {3} $ and $ \Pi _ {4} $ and a curve $ \Pi _ {5} $. Parametric formulas have been derived for $ \Pi _ {1} $ and $ \Pi _ {2} $ in terms of elementary functions. The intersection of $ V _ {3} $ with the plane $ \mathop{\rm Im} c _ {2} = 0 $ is symmetric about the planes $ \mathop{\rm Re} c _ {2} = 0 $ and $ \mathop{\rm Im} c _ {3} = 0 $. The intersection of $ V _ {3} $ with the plane $ \mathop{\rm Im} c _ {3} = 0 $ is symmetric about the planes $ \mathop{\rm Im} c _ {2} = 0 $ and $ \mathop{\rm Re} c _ {2} = 0 $. A function $ w = f ( z) $ corresponding to a point on $ \Pi _ {1} $ maps $ | z | < 1 $ onto the $ w $- plane cut by an analytic curve going to infinity. A function $ w = f ( z) $ corresponding to a point on $ \Pi _ {2} $ maps $ | z | < 1 $ onto the $ w $- plane cut by three analytic arcs, issuing from a finite point and inclined to one another at angles $ 2 \pi /3 $; one of these arcs lies on a straight line $ \mathop{\rm arg} w = \textrm{ const } $ and goes to infinity.

Among the other special regions that have been investigated are the following: the region of values $ \{ c _ {2} , c _ {3} \} $ in the subclass of $ S $ consisting of functions with real $ c _ {2} $ and $ c _ {3} $; the region of values $ \{ | c _ {k + 1 } |, | c _ {2k + 1 } | \} $ and $ \{ c _ {k + 1 } , c _ {2k + 1 } \} $, if $ \mathop{\rm Im} c _ {k + 1 } = \mathop{\rm Im} c _ {2k + 1 } = 0 $, on the subclass of bounded functions in $ S $ representable as

$$ f ( z) = z + \sum _ {n = 1 } ^ \infty c _ {nk + 1 } z ^ {nk + 1 } ; $$

the region of values $ \{ c _ {2} , c _ {3} \} $ on the subclass of bounded functions in $ S $; the region of values $ \{ c _ {2} , c _ {3} , c _ {4} \} $ on the subclass of functions in $ S $ with real $ c _ {2} , c _ {3} $ and $ c _ {4} $.

Sharp bounds for the coefficients, of the type $ | c _ {n} | \leq A _ {n} $, $ n \geq 2 $, have been obtained in the subclass of convex functions in $ S $ with $ A _ {n} = 1 $( cf. Convex function (of a complex variable)), in the subclass of star-like functions in $ S $ with $ A _ {n} = n $, in the subclass of odd star-like functions in $ S $ with $ A _ {n} = 1 $, $ n = 3, 5 \dots $ in the class of univalent functions having real coefficients with $ A _ {n} = n $, in the subclass of close-to-convex functions in $ S $ with $ A _ {n} = n $, and in the class $ S $ itself with $ A _ {n} = n $( cf. Bieberbach conjecture, [8]). In the class of functions

$$ f ( z) = z + \sum _ {n = 2 } ^ \infty c _ {n} z ^ {n} $$

which are regular and typically real in $ | z | < 1 $ one has the sharp bound $ | c _ {n} | \leq n $, $ n \geq 2 $, and in the class of Bieberbach–Eilenberg functions $ f ( z) = a _ {1} z + a _ {2} z ^ {2} + \dots $ one has the sharp bound $ | a _ {n} | \leq 1 $, $ n \geq 1 $.

Sharp bounds are known for the class $ \Sigma $ of functions

$$ F ( \zeta ) = \ \zeta + \sum _ {n = 0 } ^ \infty \frac{b _ {n} }{\zeta ^ {n} } $$

which are meromorphic and univalent in $ | \zeta | > 1 $; these are

$$ | b _ {1} | \leq 1,\ \ | b _ {2} | \leq { \frac{2}{3} } ,\ \ | b _ {3} | \leq { \frac{1}{2} } + e ^ {-} 6 . $$

For the subclass of star-like functions in $ \Sigma $, one has the sharp bound

$$ | b _ {n} | \leq \ \frac{2}{n + 1 } ,\ \ n \geq 1. $$

Sharp bounds are also known for other subclasses of $ S $ and $ \Sigma $( see [1][4]), and also for some classes of $ p $- valent functions and in classes of functions which are $ p $- valent in the mean (see [5]).

References

[1] G.M. Goluzin, "Geometric methods in the theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2] I.E. Bazilevich, , Mathematics in the USSR during 40 years: 1917–1957 , 1 , Moscow (1959) pp. 444–472 (In Russian)
[3] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[4] W.K. Hayman, "Coefficient problems for univalent functions and related function classes" J. London Math. Soc. , 40 : 3 (1965) pp. 385–406
[5] A.W. Goodman, "Open problems on univalent and multivalent functions" Bull. Amer. Math. Soc. , 74 : 6 (1968) pp. 1035–1050
[6] D. Phelps, "On a coefficient problem in univalent functions" Trans. Amer. Math. Soc. , 143 (1969) pp. 475–485
[7] A.C. Schaeffer, D.C. Spencer, "Coefficient regions for schlicht functions" , Amer. Math. Soc. Coll. Publ. , 35 , Amer. Math. Soc. (1950)
[8] L. de Branges, "A proof of the Bieberbach conjecture" Acta Math. , 154 : 1–2 (1985) pp. 137–152

Comments

For functions in the class $ \Sigma $, the estimates for $ | b _ {2} | $ and $ | b _ {3} | $ mentioned above are due to M. Schiffer [a1] and P.R. Garabedian and Schiffer [a2], respectively. The sharp bound for star-like functions in $ \Sigma $ is due to J. Clunie [a3] and C. Pommerenke [a4]. Standard references in English include [a5][a7].

References

[a1] M. Schiffer, "Sur un problème d'extrémum de la répresentation conforme" Bull. Soc. Math. France , 66 (1938) pp. 48–55
[a2] P.R. Garabedian, M. Schiffer, "A coefficient inequality for schlicht functions" Ann. of Math. , 61 (1955) pp. 116–136
[a3] J. Clunie, "On meromorphic schlicht functions" J. London Math. Soc. , 34 (1959) pp. 215–216
[a4] C. Pommerenke, "On meromorphic starlike functions" Pacific J. Math. , 13 (1963) pp. 221–235
[a5] W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1967)
[a6] C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[a7] P.L. Duren, "Univalent functions" , Springer (1983) pp. 258
[a8] O. Tammi, "Extremum problems for bounded univalent functions II" , Lect. notes in math. , 913 , Springer (1982)
How to Cite This Entry:
Coefficient problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coefficient_problem&oldid=46380
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article