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''in a domain''
 
''in a domain''
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946601.png" />, analytic in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946602.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946603.png" />-plane containing segments of the real axis, which is real on these segments and for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946604.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946605.png" />. A fundamental class of typically-real functions is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946607.png" /> of functions
+
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
 +
 
 +
$$
 +
f ( z)  = z +
 +
\sum _ {n = 2 } ^  \infty 
 +
c _ {n} z  ^ {n}
 +
$$
  
<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/t/t094/t094660/t0946608.png" /></td> </tr></table>
+
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
  
that are regular and typically real in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t0946609.png" /> (cf. [[#References|[1]]]). It follows from the definition of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466010.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466011.png" /> is real for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466012.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466013.png" /> contains the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466015.png" /> 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/t/t094/t094660/t09466016.png" /></td> </tr></table>
+
with real coefficients  $  c _ {n} $,
 +
that are regular and univalent in  $  | z | < 1 $(
 +
cf. [[Univalent function|Univalent function]]). If  $  f \in T $,
 +
then
  
with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466017.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466018.png" /> (cf. [[Univalent function|Univalent function]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466019.png" />, then
+
$$
 +
\phi ( z)  = \
 +
{
 +
\frac{1 - z  ^ {2} }{z}
 +
}
 +
f ( z) \in  C _ {r} ,
 +
$$
  
<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/t/t094/t094660/t09466020.png" /></td> </tr></table>
+
and, conversely, if  $  \phi \in C _ {r} $,
 +
then
  
and, conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466021.png" />, then
+
$$
 +
f ( z)  = \
 +
{
 +
\frac{z}{1 - z  ^ {2} }
 +
}
 +
\phi ( z)  \in  T,
 +
$$
  
<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/t/t094/t094660/t09466022.png" /></td> </tr></table>
+
where  $  C _ {r} $
 +
is the class of functions
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466024.png" /> is the class of functions
+
$$
 +
\phi ( z)  = 1 +
 +
\sum _ {n = 1 } ^  \infty 
 +
\alpha _ {n} z  ^ {n}
 +
$$
  
<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/t/t094/t094660/t09466025.png" /></td> </tr></table>
+
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 $.
  
that are regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466028.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466029.png" /> is real for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466030.png" />.
+
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]]]):
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466031.png" /> be the class of non-decreasing functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466033.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466034.png" />. Functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466035.png" /> can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466036.png" /> by Stieltjes integrals (cf. [[#References|[2]]]):
+
$$ \tag{1 }
 +
f ( z)  = \
 +
\int\limits _ { - } 1 ^ { 1 }
 +
s ( z, t) \
 +
d \alpha ( t),
 +
$$
  
<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/t/t094/t094660/t09466037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
s ( z, t)  = z ( 1 - 2tz + z  ^ {2} )  ^ {-} 1 ,\  \alpha ( t)  \in  M _ {1} ,
 +
$$
  
<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/t/t094/t094660/t09466038.png" /></td> </tr></table>
+
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 $:
  
in the sense that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466039.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466040.png" /> such that (1) holds and, conversely, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466041.png" /> formula (1) defines some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466042.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466043.png" /> for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466044.png" />. The largest domain in which every function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466045.png" /> is univalent is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466046.png" />. From the representation (1) for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466047.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466048.png" />:
+
$$ \tag{2 }
 +
- n  \leq  c _ {n}  \leq  n \ \
 +
\textrm{ if }  n  \textrm{ is }  \textrm{ even } ,
 +
$$
  
<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/t/t094/t094660/t09466049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{3 }
 +
k _ {n}  = \min _ {0 \leq  \theta \leq  \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/t/t094/t094660/t09466050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{\sin  n \theta }{\sin  \theta }
 +
  \leq  c _ {n}  \leq  n \  \textrm{ if }  n  \textrm{ is }  \textrm{ odd }
 +
$$
  
<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/t/t094/t094660/t09466051.png" /></td> </tr></table>
+
$$
 +
\left ( k _ {n}  \sim 
 +
\frac{- 2n }{3 \pi }
 +
\right ) ,
 +
$$
  
with equality on the left in (2) only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466052.png" /> and on the right only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466053.png" />, on the left in (3) only for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466054.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466055.png" />, and on the right only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466057.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466058.png" />, the coefficient regions for the systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094660/t09466062.png" />, have been found (cf. [[#References|[3]]]).
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:27, 6 June 2020


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. [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). 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. [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; 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. [3]).

References

[1] W. Rogosinski, "Ueber positive harmonische Entwicklungen und typische-reelle Potenzreihen" Math. Z. , 35 (1932) pp. 93–121
[2] G.M. Goluzin, "On typically real functions" Mat. Sb. , 27 : 2 (1950) pp. 201–218 (In Russian)
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)

Comments

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a2] A.W. Goodman, "Univalent functions" , 1 , Mariner (1983)
How to Cite This Entry:
Typically-real function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Typically-real_function&oldid=49058
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article