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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200901.png" /> be the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200902.png" /> that are analytic in the open unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200903.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200905.png" /> (cf. also [[Analytic function|Analytic function]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200906.png" /> denote the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200907.png" /> consisting of all univalent functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200908.png" /> (cf. also [[Univalent function|Univalent function]]). Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b1200909.png" /> denote the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009010.png" /> consisting of functions that are starlike with respect to the origin (cf. also [[Univalent function|Univalent function]]).
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Let $A$ be the class of functions $f ( z )$ that are analytic in the open unit disc $U$ with $f ( 0 ) = 0$ and $f ^ { \prime } ( 0 ) = 1$ (cf. also [[Analytic function|Analytic function]]). Let $S$ denote the subclass of $A$ consisting of all univalent functions in $U$ (cf. also [[Univalent function|Univalent function]]). Further, let $S ^ { * }$ denote the subclass of $S$ consisting of functions that are starlike with respect to the origin (cf. also [[Univalent function|Univalent function]]).
  
 
The Kufarev differential equation
 
The Kufarev differential equation
  
<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/b/b120/b120090/b12009011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} \frac { \partial f ( z , t ) } { \partial t } = - f ( z , t ) p ( f , t ), \end{equation}
  
<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/b/b120/b120090/b12009012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \frac { \partial f ( z , t ) } { \partial t } = - z f ^ { \prime } ( z , t ) p ( z , t ), \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009013.png" /> is a function regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009014.png" />, having positive real part and being piecewise continuous with respect to a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009015.png" />, plays an important part in the theory of univalent functions. This differential equation can be generalized as the corresponding Loewner differential equation
+
where $p ( u , t ) = 1 + \alpha _ { 1 } ( t ) u + \alpha _ { 2 } ( t ) u ^ { 2 } +\dots$ is a function regular in $| u | < 1$, having positive real part and being piecewise continuous with respect to a parameter $t$, plays an important part in the theory of univalent functions. This differential equation can be generalized as the corresponding Loewner differential equation
  
<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/b/b120/b120090/b12009016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \frac { \partial f ( z , t ) } { \partial t } = - f ( z , t ) \frac { 1 + k f ( z , t ) } { 1 - k  f ( z , t ) }, \end{equation}
  
<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/b/b120/b120090/b12009017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} \frac { \partial f ( z , t ) } { \partial t } = - z f ^ { \prime } ( z , t ) \frac { 1 + k z } { 1 - k z }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009018.png" /> is a continuous complex-valued function with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009020.png" />).
+
where $ k  = k ( t )$ is a continuous complex-valued function with $| k ( t ) | = 1$ ($0 \leq t < \infty$).
  
Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009022.png" />), (a1) can be written in the form
+
Letting $\tau = e ^ { - t }$ ($0 < \tau \leq 1$), (a1) can be written in the form
  
<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/b/b120/b120090/b12009023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a5} \frac { d \tau } { \tau } = p ( f , \tau ) \frac { d f } { f }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009026.png" /> is a function regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009027.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009028.png" />. Introducing a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009029.png" />, one sets
+
where $f = f ( z , \tau )$, $f ( z , 1 ) = z$, and $p ( f , \tau ) = 1 + \alpha _ { 1 } ( \tau ) f + \alpha _ { 2 } ( \tau ) f ^ { 2 } +\dots $ is a function regular in $| f | < 1$ with $\operatorname { Re } p ( f , \tau ) > 0$. Introducing a real parameter $a$, one sets
  
<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/b/b120/b120090/b12009030.png" /></td> </tr></table>
+
\begin{equation*} p _ { 1 } (\, f , \tau ) = p ( e ^ { i a \text{ln} \tau } f , \tau ). \end{equation*}
  
Further, making the change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009031.png" />, one obtains
+
Further, making the change $\xi = e ^ { i a\operatorname{ln} \tau } f$, one obtains
  
<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/b/b120/b120090/b12009032.png" /></td> </tr></table>
+
\begin{equation*} \frac { d f } { f } = \frac { d \xi } { \xi } - i a \frac { d \tau } { \tau }. \end{equation*}
  
Making the changes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009033.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009034.png" /> and
+
Making the changes $1 / p ( \xi , \tau ) = p _ { 2 } ( \xi , \tau )$ with $\operatorname { Re } p _ { 2 } ( \xi , \tau ) > 0$ 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/b/b120/b120090/b12009035.png" /></td> </tr></table>
+
\begin{equation*} \frac { 1 } { p _ { 2 } ( \xi , \tau ) + a i } = \frac { p _ { 3 } ( \xi , \tau ) } { 1 + a ^ { 2 } } - \frac { a i } { 1 + a ^ { 2 } } \end{equation*}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009036.png" />, one obtains
+
with $\operatorname { Re } p _ { 3 } ( \xi , \tau ) > 0$, one obtains
  
<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/b/b120/b120090/b12009037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\begin{equation} \tag{a6} ( 1 + a ^ { 2 } ) \frac { d \tau } { \tau } = ( p _ { 3 } ( \xi , \tau ) - a i ) \frac { d \xi } { \xi }, \end{equation}
  
 
which is the generalization of (a5).
 
which is the generalization of (a5).
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Writing
 
Writing
  
<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/b/b120/b120090/b12009038.png" /></td> </tr></table>
+
\begin{equation*} p _ { 3 } ( \xi , \tau ) = p _ { 0 } ( \xi ) ( 1 - \tau ^ { m } ) + p _ { 1 } ( \xi ) \tau ^ { m }\; ( m > 0 ) \end{equation*}
  
 
with
 
with
  
<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/b/b120/b120090/b12009039.png" /></td> </tr></table>
+
\begin{equation*} p _ { 0 } ( \xi ) = 1 + \alpha _ { 1 } \xi + \alpha _ { 2 } \xi ^ { 2 } + \ldots ( \operatorname { Re } p _ { 0 } ( \xi ) > 0 ) \end{equation*}
  
 
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/b/b120/b120090/b12009040.png" /></td> </tr></table>
+
\begin{equation*} p _ { 1 } ( \xi ) = 1 + \beta _ { 1 } \xi + \beta _ { 2 } \xi ^ { 2 } + \ldots ( \operatorname { Re } p _ { 1 } ( \xi ) > 0 ), \end{equation*}
  
 
(a6) gives the Bernoulli equation
 
(a6) gives the Bernoulli equation
  
<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/b/b120/b120090/b12009041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
\begin{equation} \tag{a7} ( 1 + a ^ { 2 } ) \frac { d \tau } { d \xi } = \end{equation}
  
<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/b/b120/b120090/b12009042.png" /></td> </tr></table>
+
\begin{equation*} = ( p _ { 0 } ( \xi ) - a i ) \frac { \tau } { \xi } + ( p _ { 1 } ( \xi ) + p _ { 0 } ( \xi ) ) \frac { \tau ^ { m + 1 } } { \xi }. \end{equation*}
  
 
If one takes
 
If one takes
  
<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/b/b120/b120090/b12009043.png" /></td> </tr></table>
+
\begin{equation*} \xi = e ^ { i a \operatorname { ln } \tau } f ( z , \tau ) | _ { \tau = 1 } = z \end{equation*}
  
 
in (a7), one obtains the integral
 
in (a7), one obtains the integral
  
<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/b/b120/b120090/b12009044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
\begin{equation} \tag{a8} - \frac { 1 + a ^ { 2 } } { m } \tau ^ { - m } = \end{equation}
  
<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/b/b120/b120090/b12009045.png" /></td> </tr></table>
+
\begin{equation*} =e ^{-\frac { m } { 1 + a ^ { 2 } } \int _ { z } ^ { \xi } \frac { p _ { 0 } ( s ) - a i } { s } d s}  \times \times \left\{ \int _ { z } ^ { \xi } \frac { p _ { 1 } ( s ) - p _ { 0 } ( s ) } { s } e^{ \frac { m } { 1 + a ^ { 2 } } \int _ { z } ^ { s } \frac { p _ { 0 } ( t ) - a i } { t } d t  } d s - \frac { 1 + a ^ { 2 } } { m } \right\}. \end{equation*}
  
 
Using
 
Using
  
<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/b/b120/b120090/b12009046.png" /></td> </tr></table>
+
\begin{equation*} \int _ { z } ^ { \xi } \frac { 1 - a i } { s } d s = \operatorname { ln } \left( \frac { \xi } { z } \right) ^ { 1 - a i } \end{equation*}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009047.png" />, one sees that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009048.png" /> is uniformly convergent to a certain function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009049.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009050.png" />. This implies that
+
and $f e ^ { i a \operatorname { ln } \tau } = f e ^ { a i } = \xi$, one sees that $f ( z , \tau ) / \tau$ is uniformly convergent to a certain function $w = w ( z )$ of the class $S$. This implies that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\begin{equation} \tag{a9} w ^ { \frac { m } { 1 + a i } } = \end{equation}
  
<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/b/b120/b120090/b12009052.png" /></td> </tr></table>
+
\begin{equation*} =\frac { m } { 1 + a ^ { 2 } } \left\{ \int _ { 0 } ^ { z } \frac { p _ { 1 } ( s ) - p _ { 0 } ( s ) } { s ^ { 1 - \frac { m } { 1 + a i } } } e ^ { \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { s } \frac { p _ { 0 } ( t ) - 1 } { t } d t } d s + + \frac { 1 + a ^ { 2 } } { m } z ^ { \frac { m } { 1 + a i } } e ^ { \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { z } \frac { p _ { 0 } ( t ) - 1 } { t } d t} \right\}. \end{equation*}
  
 
Noting that (a9) implies
 
Noting that (a9) implies
  
<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/b/b120/b120090/b12009053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
+
\begin{equation} \tag{a10} w ( z ) = \end{equation}
  
<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/b/b120/b120090/b12009054.png" /></td> </tr></table>
+
\begin{equation*} = \left\{ \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { z } \frac { p _ { 1 } ( s ) - a i } { s ^ { 1 - \frac { m } { 1 + a i } } } e ^ { \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { s } \frac { p _ { 0 } ( t ) - 1 } { t } d t } d s \right\}^{\frac{1+ai}{m}}, \end{equation*}
  
I.E. Bazilevich [[#References|[a1]]] proved that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009055.png" /> given by
+
I.E. Bazilevich [[#References|[a1]]] proved that the function $f ( z )$ given by
  
<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/b/b120/b120090/b12009056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
+
\begin{equation} \tag{a11} f ( z ) = \end{equation}
  
<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/b/b120/b120090/b12009057.png" /></td> </tr></table>
+
\begin{equation*} = \left\{ \frac { \beta } { 1 + \alpha ^ { 2 } } \int _ { 0 } ^ { z } \frac { h ( \xi ) - \alpha i } { \xi ^ { 1 + \alpha \beta i / ( 1 + \alpha ^ { 2 } ) } } g ( \xi ) ^ { \beta / ( 1 + \alpha ^ { 2 } ) } d \xi \right\} ^ { ( 1 + \alpha i ) / \beta } \end{equation*}
  
belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009058.png" />, where
+
belongs to the class $S$, where
  
<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/b/b120/b120090/b12009059.png" /></td> </tr></table>
+
\begin{equation*} g ( z ) = z e ^ { \int _ { 0 } ^ { z } \frac { p _ { 0 } ( t ) - 1 } { t } d t }  \in S^*, \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009060.png" /> is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009061.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009063.png" /> is any real number, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009064.png" />.
+
$h ( z ) = 1 + c _ { 1 } z + c _ { 2 } z ^ { 2 } + \ldots$ is regular in $U$ with $\operatorname { Re } h ( z ) > 0$, $\alpha$ is any real number, and $\beta > 0$.
  
If one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009065.png" /> in (a11), then
+
If one sets $\alpha = 0$ in (a11), then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
+
\begin{equation} \tag{a12} f ( z ) = \left( \beta \int _ { 0 } ^ { z } h ( \xi ) \xi ^ { - 1 } g ( \xi ) ^ { \beta } d \xi \right) ^ { 1 / \beta }. \end{equation}
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009067.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009068.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009069.png" /> given by (a12) satisfies
+
Since $\operatorname { Re } h ( z ) > 0$ in $U$, the function $f ( z )$ given by (a12) satisfies
  
<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/b/b120/b120090/b12009070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
+
\begin{equation} \tag{a13} \operatorname { Re } \left\{ \frac { z f ^ { \prime } ( z ) } { f ( z ) ^ { 1 - \beta } g ( z ) ^ { \beta } } \right\} > 0 ( z \in U ). \end{equation}
  
Therefore, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009071.png" /> satisfying (a13) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009072.png" /> is called a Bazilevich function of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009074.png" />.
+
Therefore, the function $f ( z )$ satisfying (a13) with $g ( z ) \in S ^ { * }$ is called a Bazilevich function of type $\beta$.
  
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009075.png" /> the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009076.png" /> that are Bazilevich of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009077.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009078.png" />.
+
Denote by $B ( \beta )$ the class of functions $f ( z )$ that are Bazilevich of type $\beta$ in $U$.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009079.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009080.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009081.png" />, then
+
1) If $f \in B ( \beta )$ with $| f ( z ) | < 1$ in $U$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009082.png" /></td> </tr></table>
+
\begin{equation*} L ( r ) = \int _ { 0 } ^ { 2 \pi } \left| z f ^ { \prime } ( z ) \right| d \theta = O \left( \operatorname { log } \frac { 1 } { 1 - r } \right) \end{equation*}
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009083.png" /> (see [[#References|[a3]]]).
+
as $r \rightarrow 1$ (see [[#References|[a3]]]).
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009084.png" /> be analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009085.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009086.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009088.png" /> (see [[#References|[a7]]]).
+
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009084.png"/> be analytic in $U$. Then $\varphi ( z ) \in B ( \beta )$ if and only if $\varphi ( z ) = ( f ( z ^ { m } ) ) ^ { 1 / m }$ with $f ( z ) \in B ( \alpha / m )$ (see [[#References|[a7]]]).
  
3) T. Sheil-Small [[#References|[a12]]] has introduced the class of Bazilevich functions of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009090.png" />, given by
+
3) T. Sheil-Small [[#References|[a12]]] has introduced the class of Bazilevich functions of type $( \alpha , \beta )$, given by
  
<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/b/b120/b120090/b12009091.png" /></td> </tr></table>
+
\begin{equation*} f ( z ) = \{ \int _ { 0 } ^ { z } g ^ { \alpha } ( \xi ) h ( \xi ) \xi ^ { i \beta - 1 } d \xi \} ^ { 1 / ( \alpha + i \beta ) }. \end{equation*}
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009092.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009093.png" /> is a close-to-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009094.png" />-valent function, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120090/b12009095.png" /> is a rational number (see [[#References|[a9]]]).
+
4) If $f \in B ( m / n )$, then $( f ( z ^ { n } ) )^ { m / n }$ is a close-to-convex $m$-valent function, where $m / n$ is a rational number (see [[#References|[a9]]]).
  
 
For other properties of Bazilevich functions, see [[#References|[a4]]], [[#References|[a8]]], [[#References|[a10]]], [[#References|[a2]]], [[#References|[a6]]], [[#References|[a11]]], and [[#References|[a5]]].
 
For other properties of Bazilevich functions, see [[#References|[a4]]], [[#References|[a8]]], [[#References|[a10]]], [[#References|[a2]]], [[#References|[a6]]], [[#References|[a11]]], and [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.E. Bazilevich,  "On a class of integrability by quadratures of the equation of Loewner–Kufarev"  ''Mat. Sb.'' , '''37'''  (1955)  pp. 471–476</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Singh,  "On Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''38'''  (1973)  pp. 261–271</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.K. Thomas,  "On Bazilevič functions"  ''Trans. Amer. Math. Soc.'' , '''132'''  (1968)  pp. 353–361</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Zamorski,  "On Bazilevič schlicht functions"  ''Ann. Polon. Math.'' , '''12'''  (1962)  pp. 83–90</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , ''Grundl. Math. Wissenschaft.'' , '''259''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.J. Eenigenburg,  S.S. Miller,  P.T. Mocanu,  M.O. Reade,  "On a subclass of Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''45'''  (1974)  pp. 88–92</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F.R. Keogh,  S.S. Miller,  "On the coefficients of Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''30'''  (1971)  pp. 492–496</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  S.S. Miller,  "The Hardy class of a Bazilevič function and its derivative"  ''Proc. Amer. Math. Soc.'' , '''30'''  (1971)  pp. 125–132</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P.T. Mocanu,  M.O. Reade,  E.J. Zlotkiewicz,  "On Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''39'''  (1973)  pp. 173–174</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Nunokawa,  "On the Bazilevič analytic functions"  ''Sci. Rep. Fac. Edu. Gunma Univ.'' , '''21'''  (1972)  pp. 9–13</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Ch. Pommerenke,  "Univalent functions" , Vandenhoeck&amp;Ruprecht  (1975)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  T. Sheil-Small,  "On Bazilevič functions"  ''Quart. J. Math.'' , '''23'''  (1972)  pp. 135–142</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  I.E. Bazilevich,  "On a class of integrability by quadratures of the equation of Loewner–Kufarev"  ''Mat. Sb.'' , '''37'''  (1955)  pp. 471–476</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R. Singh,  "On Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''38'''  (1973)  pp. 261–271</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D.K. Thomas,  "On Bazilevič functions"  ''Trans. Amer. Math. Soc.'' , '''132'''  (1968)  pp. 353–361</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Zamorski,  "On Bazilevič schlicht functions"  ''Ann. Polon. Math.'' , '''12'''  (1962)  pp. 83–90</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P.L. Duren,  "Univalent functions" , ''Grundl. Math. Wissenschaft.'' , '''259''' , Springer  (1983)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  P.J. Eenigenburg,  S.S. Miller,  P.T. Mocanu,  M.O. Reade,  "On a subclass of Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''45'''  (1974)  pp. 88–92</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  F.R. Keogh,  S.S. Miller,  "On the coefficients of Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''30'''  (1971)  pp. 492–496</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  S.S. Miller,  "The Hardy class of a Bazilevič function and its derivative"  ''Proc. Amer. Math. Soc.'' , '''30'''  (1971)  pp. 125–132</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  P.T. Mocanu,  M.O. Reade,  E.J. Zlotkiewicz,  "On Bazilevič functions"  ''Proc. Amer. Math. Soc.'' , '''39'''  (1973)  pp. 173–174</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  M. Nunokawa,  "On the Bazilevič analytic functions"  ''Sci. Rep. Fac. Edu. Gunma Univ.'' , '''21'''  (1972)  pp. 9–13</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  Ch. Pommerenke,  "Univalent functions" , Vandenhoeck&amp;Ruprecht  (1975)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  T. Sheil-Small,  "On Bazilevič functions"  ''Quart. J. Math.'' , '''23'''  (1972)  pp. 135–142</td></tr></table>

Latest revision as of 18:54, 24 January 2024

Let $A$ be the class of functions $f ( z )$ that are analytic in the open unit disc $U$ with $f ( 0 ) = 0$ and $f ^ { \prime } ( 0 ) = 1$ (cf. also Analytic function). Let $S$ denote the subclass of $A$ consisting of all univalent functions in $U$ (cf. also Univalent function). Further, let $S ^ { * }$ denote the subclass of $S$ consisting of functions that are starlike with respect to the origin (cf. also Univalent function).

The Kufarev differential equation

\begin{equation} \tag{a1} \frac { \partial f ( z , t ) } { \partial t } = - f ( z , t ) p ( f , t ), \end{equation}

\begin{equation} \tag{a2} \frac { \partial f ( z , t ) } { \partial t } = - z f ^ { \prime } ( z , t ) p ( z , t ), \end{equation}

where $p ( u , t ) = 1 + \alpha _ { 1 } ( t ) u + \alpha _ { 2 } ( t ) u ^ { 2 } +\dots$ is a function regular in $| u | < 1$, having positive real part and being piecewise continuous with respect to a parameter $t$, plays an important part in the theory of univalent functions. This differential equation can be generalized as the corresponding Loewner differential equation

\begin{equation} \tag{a3} \frac { \partial f ( z , t ) } { \partial t } = - f ( z , t ) \frac { 1 + k f ( z , t ) } { 1 - k f ( z , t ) }, \end{equation}

\begin{equation} \tag{a4} \frac { \partial f ( z , t ) } { \partial t } = - z f ^ { \prime } ( z , t ) \frac { 1 + k z } { 1 - k z }, \end{equation}

where $ k = k ( t )$ is a continuous complex-valued function with $| k ( t ) | = 1$ ($0 \leq t < \infty$).

Letting $\tau = e ^ { - t }$ ($0 < \tau \leq 1$), (a1) can be written in the form

\begin{equation} \tag{a5} \frac { d \tau } { \tau } = p ( f , \tau ) \frac { d f } { f }, \end{equation}

where $f = f ( z , \tau )$, $f ( z , 1 ) = z$, and $p ( f , \tau ) = 1 + \alpha _ { 1 } ( \tau ) f + \alpha _ { 2 } ( \tau ) f ^ { 2 } +\dots $ is a function regular in $| f | < 1$ with $\operatorname { Re } p ( f , \tau ) > 0$. Introducing a real parameter $a$, one sets

\begin{equation*} p _ { 1 } (\, f , \tau ) = p ( e ^ { i a \text{ln} \tau } f , \tau ). \end{equation*}

Further, making the change $\xi = e ^ { i a\operatorname{ln} \tau } f$, one obtains

\begin{equation*} \frac { d f } { f } = \frac { d \xi } { \xi } - i a \frac { d \tau } { \tau }. \end{equation*}

Making the changes $1 / p ( \xi , \tau ) = p _ { 2 } ( \xi , \tau )$ with $\operatorname { Re } p _ { 2 } ( \xi , \tau ) > 0$ and

\begin{equation*} \frac { 1 } { p _ { 2 } ( \xi , \tau ) + a i } = \frac { p _ { 3 } ( \xi , \tau ) } { 1 + a ^ { 2 } } - \frac { a i } { 1 + a ^ { 2 } } \end{equation*}

with $\operatorname { Re } p _ { 3 } ( \xi , \tau ) > 0$, one obtains

\begin{equation} \tag{a6} ( 1 + a ^ { 2 } ) \frac { d \tau } { \tau } = ( p _ { 3 } ( \xi , \tau ) - a i ) \frac { d \xi } { \xi }, \end{equation}

which is the generalization of (a5).

Writing

\begin{equation*} p _ { 3 } ( \xi , \tau ) = p _ { 0 } ( \xi ) ( 1 - \tau ^ { m } ) + p _ { 1 } ( \xi ) \tau ^ { m }\; ( m > 0 ) \end{equation*}

with

\begin{equation*} p _ { 0 } ( \xi ) = 1 + \alpha _ { 1 } \xi + \alpha _ { 2 } \xi ^ { 2 } + \ldots ( \operatorname { Re } p _ { 0 } ( \xi ) > 0 ) \end{equation*}

and

\begin{equation*} p _ { 1 } ( \xi ) = 1 + \beta _ { 1 } \xi + \beta _ { 2 } \xi ^ { 2 } + \ldots ( \operatorname { Re } p _ { 1 } ( \xi ) > 0 ), \end{equation*}

(a6) gives the Bernoulli equation

\begin{equation} \tag{a7} ( 1 + a ^ { 2 } ) \frac { d \tau } { d \xi } = \end{equation}

\begin{equation*} = ( p _ { 0 } ( \xi ) - a i ) \frac { \tau } { \xi } + ( p _ { 1 } ( \xi ) + p _ { 0 } ( \xi ) ) \frac { \tau ^ { m + 1 } } { \xi }. \end{equation*}

If one takes

\begin{equation*} \xi = e ^ { i a \operatorname { ln } \tau } f ( z , \tau ) | _ { \tau = 1 } = z \end{equation*}

in (a7), one obtains the integral

\begin{equation} \tag{a8} - \frac { 1 + a ^ { 2 } } { m } \tau ^ { - m } = \end{equation}

\begin{equation*} =e ^{-\frac { m } { 1 + a ^ { 2 } } \int _ { z } ^ { \xi } \frac { p _ { 0 } ( s ) - a i } { s } d s} \times \times \left\{ \int _ { z } ^ { \xi } \frac { p _ { 1 } ( s ) - p _ { 0 } ( s ) } { s } e^{ \frac { m } { 1 + a ^ { 2 } } \int _ { z } ^ { s } \frac { p _ { 0 } ( t ) - a i } { t } d t } d s - \frac { 1 + a ^ { 2 } } { m } \right\}. \end{equation*}

Using

\begin{equation*} \int _ { z } ^ { \xi } \frac { 1 - a i } { s } d s = \operatorname { ln } \left( \frac { \xi } { z } \right) ^ { 1 - a i } \end{equation*}

and $f e ^ { i a \operatorname { ln } \tau } = f e ^ { a i } = \xi$, one sees that $f ( z , \tau ) / \tau$ is uniformly convergent to a certain function $w = w ( z )$ of the class $S$. This implies that

\begin{equation} \tag{a9} w ^ { \frac { m } { 1 + a i } } = \end{equation}

\begin{equation*} =\frac { m } { 1 + a ^ { 2 } } \left\{ \int _ { 0 } ^ { z } \frac { p _ { 1 } ( s ) - p _ { 0 } ( s ) } { s ^ { 1 - \frac { m } { 1 + a i } } } e ^ { \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { s } \frac { p _ { 0 } ( t ) - 1 } { t } d t } d s + + \frac { 1 + a ^ { 2 } } { m } z ^ { \frac { m } { 1 + a i } } e ^ { \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { z } \frac { p _ { 0 } ( t ) - 1 } { t } d t} \right\}. \end{equation*}

Noting that (a9) implies

\begin{equation} \tag{a10} w ( z ) = \end{equation}

\begin{equation*} = \left\{ \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { z } \frac { p _ { 1 } ( s ) - a i } { s ^ { 1 - \frac { m } { 1 + a i } } } e ^ { \frac { m } { 1 + a ^ { 2 } } \int _ { 0 } ^ { s } \frac { p _ { 0 } ( t ) - 1 } { t } d t } d s \right\}^{\frac{1+ai}{m}}, \end{equation*}

I.E. Bazilevich [a1] proved that the function $f ( z )$ given by

\begin{equation} \tag{a11} f ( z ) = \end{equation}

\begin{equation*} = \left\{ \frac { \beta } { 1 + \alpha ^ { 2 } } \int _ { 0 } ^ { z } \frac { h ( \xi ) - \alpha i } { \xi ^ { 1 + \alpha \beta i / ( 1 + \alpha ^ { 2 } ) } } g ( \xi ) ^ { \beta / ( 1 + \alpha ^ { 2 } ) } d \xi \right\} ^ { ( 1 + \alpha i ) / \beta } \end{equation*}

belongs to the class $S$, where

\begin{equation*} g ( z ) = z e ^ { \int _ { 0 } ^ { z } \frac { p _ { 0 } ( t ) - 1 } { t } d t } \in S^*, \end{equation*}

$h ( z ) = 1 + c _ { 1 } z + c _ { 2 } z ^ { 2 } + \ldots$ is regular in $U$ with $\operatorname { Re } h ( z ) > 0$, $\alpha$ is any real number, and $\beta > 0$.

If one sets $\alpha = 0$ in (a11), then

\begin{equation} \tag{a12} f ( z ) = \left( \beta \int _ { 0 } ^ { z } h ( \xi ) \xi ^ { - 1 } g ( \xi ) ^ { \beta } d \xi \right) ^ { 1 / \beta }. \end{equation}

Since $\operatorname { Re } h ( z ) > 0$ in $U$, the function $f ( z )$ given by (a12) satisfies

\begin{equation} \tag{a13} \operatorname { Re } \left\{ \frac { z f ^ { \prime } ( z ) } { f ( z ) ^ { 1 - \beta } g ( z ) ^ { \beta } } \right\} > 0 ( z \in U ). \end{equation}

Therefore, the function $f ( z )$ satisfying (a13) with $g ( z ) \in S ^ { * }$ is called a Bazilevich function of type $\beta$.

Denote by $B ( \beta )$ the class of functions $f ( z )$ that are Bazilevich of type $\beta$ in $U$.

1) If $f \in B ( \beta )$ with $| f ( z ) | < 1$ in $U$, then

\begin{equation*} L ( r ) = \int _ { 0 } ^ { 2 \pi } \left| z f ^ { \prime } ( z ) \right| d \theta = O \left( \operatorname { log } \frac { 1 } { 1 - r } \right) \end{equation*}

as $r \rightarrow 1$ (see [a3]).

2) Let be analytic in $U$. Then $\varphi ( z ) \in B ( \beta )$ if and only if $\varphi ( z ) = ( f ( z ^ { m } ) ) ^ { 1 / m }$ with $f ( z ) \in B ( \alpha / m )$ (see [a7]).

3) T. Sheil-Small [a12] has introduced the class of Bazilevich functions of type $( \alpha , \beta )$, given by

\begin{equation*} f ( z ) = \{ \int _ { 0 } ^ { z } g ^ { \alpha } ( \xi ) h ( \xi ) \xi ^ { i \beta - 1 } d \xi \} ^ { 1 / ( \alpha + i \beta ) }. \end{equation*}

4) If $f \in B ( m / n )$, then $( f ( z ^ { n } ) )^ { m / n }$ is a close-to-convex $m$-valent function, where $m / n$ is a rational number (see [a9]).

For other properties of Bazilevich functions, see [a4], [a8], [a10], [a2], [a6], [a11], and [a5].

References

[a1] I.E. Bazilevich, "On a class of integrability by quadratures of the equation of Loewner–Kufarev" Mat. Sb. , 37 (1955) pp. 471–476
[a2] R. Singh, "On Bazilevič functions" Proc. Amer. Math. Soc. , 38 (1973) pp. 261–271
[a3] D.K. Thomas, "On Bazilevič functions" Trans. Amer. Math. Soc. , 132 (1968) pp. 353–361
[a4] J. Zamorski, "On Bazilevič schlicht functions" Ann. Polon. Math. , 12 (1962) pp. 83–90
[a5] P.L. Duren, "Univalent functions" , Grundl. Math. Wissenschaft. , 259 , Springer (1983)
[a6] P.J. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, "On a subclass of Bazilevič functions" Proc. Amer. Math. Soc. , 45 (1974) pp. 88–92
[a7] F.R. Keogh, S.S. Miller, "On the coefficients of Bazilevič functions" Proc. Amer. Math. Soc. , 30 (1971) pp. 492–496
[a8] S.S. Miller, "The Hardy class of a Bazilevič function and its derivative" Proc. Amer. Math. Soc. , 30 (1971) pp. 125–132
[a9] P.T. Mocanu, M.O. Reade, E.J. Zlotkiewicz, "On Bazilevič functions" Proc. Amer. Math. Soc. , 39 (1973) pp. 173–174
[a10] M. Nunokawa, "On the Bazilevič analytic functions" Sci. Rep. Fac. Edu. Gunma Univ. , 21 (1972) pp. 9–13
[a11] Ch. Pommerenke, "Univalent functions" , Vandenhoeck&Ruprecht (1975)
[a12] T. Sheil-Small, "On Bazilevič functions" Quart. J. Math. , 23 (1972) pp. 135–142
How to Cite This Entry:
Bazilevich functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bazilevich_functions&oldid=15825
This article was adapted from an original article by S. Owa (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article