View source for Typically-real function

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''in a domain''

A function  $  f ( z) $, 
analytic in some domain  $  B $
in the  $  z $-
plane containing segments of the real axis, which is real on these segments and for which  $  (  \mathop{\rm Im}  f ( z)) (  \mathop{\rm Im}  z) > 0 $
whenever  $   \mathop{\rm Im}  z \neq 0 $. 
A fundamental class of typically-real functions is the class  $  T $
of functions

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

that are regular and typically real in the disc  $  | z | < 1 $(
cf. [[#References|[1]]]). It follows from the definition of the class  $  T $
that  $  c _ {n} $
is real for  $  n \geq  2 $. 
The class  $  T $
contains the class  $  S _ {r} $
of functions

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

with real coefficients  $  c _ {n} $, 
that are regular and univalent in  $  | z | < 1 $(
cf. [[Univalent function|Univalent function]]). If  $  f \in T $, 
then

$$ 
\phi ( z)  = \ 
{
\frac{1 - z  ^ {2} }{z}
 }
f ( z)  \in  C _ {r} ,
$$

and, conversely, if  $  \phi \in C _ {r} $, 
then

$$ 
f ( z)  = \ 
{
\frac{z}{1 - z  ^ {2} }
 }
\phi ( z)  \in  T,
$$

where  $  C _ {r} $
is the class of functions

$$ 
\phi ( z)  =  1 +
\sum _ {n = 1 } ^  \infty  
\alpha _ {n} z  ^ {n}
$$

that are regular in  $  | z | < 1 $
with  $   \mathop{\rm Re}  \phi ( z) > 0 $
in  $  | z | < 1 $
and such that  $  \alpha _ {n} $
is real for  $  n \geq  1 $.

Let  $  M _ {1} $
be the class of non-decreasing functions  $  \alpha ( t) $
on  $  [- 1, 1] $
for which  $  \alpha ( 1) - \alpha (- 1) = 1 $. 
Functions of class  $  T $
can be represented in  $  | z | < 1 $
by Stieltjes integrals (cf. [[#References|[2]]]):

$$ \tag{1 }
f ( z)  = \ 
\int\limits _ { - } 1 ^ { 1 } 
s ( z, t) \ 
d \alpha ( t),
$$

$$ 
s ( z, t)  =  z ( 1 - 2tz + z  ^ {2} )  ^ {-} 1 ,\  \alpha ( t)  \in  M _ {1} ,
$$

in the sense that for each  $  f \in T $
there exists an  $  \alpha \in M _ {1} $
such that (1) holds and, conversely, for any  $  \alpha \in M _ {1} $
formula (1) defines some function  $  f \in T $. 
One has  $  s ( z, t) \in S _ {r} $
for any fixed  $  t \in [- 1, 1] $. 
The largest domain in which every function in  $  T $
is univalent is  $  \{ | z + i | < \sqrt 2 \} \cap \{ | z - i | < \sqrt 2 \} $. 
From the representation (1) for the class  $  T $, 
a number of rotation and distortion theorems have been obtained (cf. [[Distortion theorems|Distortion theorems]]; [[Rotation theorems|Rotation theorems]]). The following hold in the class  $  T $:

$$ \tag{2 }
- n  \leq   c _ {n}  \leq   n \ \ 
\textrm{ if }  n  \textrm{ is }  \textrm{ even } ,
$$

$$ \tag{3 }
k _ {n}  =  \min _ {0 \leq  \theta \leq  \pi } \ 

\frac{\sin  n \theta }{\sin  \theta }
  \leq   c _ {n}  \leq   n \  \textrm{ if }  n  \textrm{ is }  \textrm{ odd }
$$

$$ 
\left ( k _ {n}  \sim  
\frac{- 2n }{3 \pi }
 \right ) ,
$$

with equality on the left in (2) only for  $  s ( z, - 1) $
and on the right only for  $  s ( z, 1) $, 
on the left in (3) only for functions  $  f ( z) = \lambda s ( z, t _ {n} ) + ( 1 - \lambda ) s ( z, - t _ {n} ) $
for some  $  t _ {n} \in [- 1, 1] $, 
and on the right only for  $  f ( z) = \lambda s ( z, 1) + ( 1 - \lambda ) s ( z, - 1) $, 
$  0 \leq  \lambda \leq  1 $.

For  $  T $, 
the coefficient regions for the systems  $  \{ c _ {2} \dots c _ {n} \} $, 
$  \{ f ( z) \} $, 
$  \{ f ( z), c _ {2} \dots c _ {n} \} $, 
$  n \geq  2 $, 
have been found (cf. [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Rogosinski,   "Ueber positive harmonische Entwicklungen und typische-reelle Potenzreihen"  ''Math. Z.'' , '''35'''  (1932)  pp. 93–121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,   "On typically real functions"  ''Mat. Sb.'' , '''27''' :  2  (1950)  pp. 201–218  (In Russian)</TD></TR><TR><TD valign="top">[3]</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></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.W. Goodman,   "Univalent functions" , '''1''' , Mariner  (1983)</TD></TR></table>