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''univalent star-like function''
 
''univalent star-like function''
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872201.png" /> which is regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872203.png" />, and maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872204.png" /> onto a [[Star-like domain|star-like domain]] with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872205.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872207.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722010.png" />, regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722011.png" />, is star-like in this disc if and only if it satisfies the condition
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A function $w=f(z)$ which is regular and univalent in the disc $|z|<1$, $f(0)=0$, and maps $|z|<1$ onto a [[Star-like domain|star-like domain]] with respect to $w=0$. A function $f(z)$, $f(z)\neq0$ in $0<|z|<1$, $f(0)=0$, $f'(0)\neq0$, regular in $|z|<1$, is star-like in this disc if and only if it satisfies the condition
  
<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/s/s087/s087220/s08722012.png" /></td> </tr></table>
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$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>0,\quad|z|<1.$$
  
The family of star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722013.png" />, normalized so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722015.png" />, forms the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722017.png" />, which admits a parametric representation by Stieltjes integrals:
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The family of star-like functions in $|z|<1$, normalized so that $f(0)=0$, $f'(0)=1$, forms the class $S^*$, which admits a parametric representation by Stieltjes integrals:
  
<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/s/s087/s087220/s08722018.png" /></td> </tr></table>
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$$f(z)=z\exp\left[-2\int_{-\pi}^{\pi}\log(1-e^{-it}z)d\mu(t)\right],$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722019.png" /> is a non-decreasing function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722021.png" />.
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where $\mu(t)$ is a non-decreasing function on $[-\pi,\pi]$, $\mu(\pi)-\mu(-\pi)=1$.
  
For the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722022.png" /> the [[Coefficient problem|coefficient problem]] has been solved; sharp estimates have been found for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722026.png" /> (the argument of the function is the branch that vanishes at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722027.png" />). The extremal functions for these estimates are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722029.png" /> is real. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722030.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722031.png" /> is related to the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722034.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722035.png" /> and map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722036.png" /> onto a convex domain, by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722037.png" />.
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For the class $S^*$ the [[Coefficient problem|coefficient problem]] has been solved; sharp estimates have been found for $|f(z)|$, $|f'(z)|$, $\arg f(z)$, $\arg f'(z)$ (the argument of the function is the branch that vanishes at $z=0$). The extremal functions for these estimates are $f(z)=z/(1-e^{i\theta}z)^2$, where $\theta$ is real. The class $S^*$ of functions $f(z)$ is related to the class of functions $\phi(z)$, $\phi(0)=0$, $\phi'(0)=1$, that are regular and univalent in $|z|<1$ and map $|z|<1$ onto a convex domain, by the formula $z\phi'(z)=f(z)$.
  
 
A star-like function such that
 
A star-like function such 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/s/s087/s087220/s08722038.png" /></td> </tr></table>
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$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>\alpha,\quad|z|,\alpha<1,$$
  
is called a star-like function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722041.png" />.
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is called a star-like function of order $\alpha$ in $|z|<1$.
  
Attention has also been given to univalent star-like functions in an annulus (see [[#References|[1]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722042.png" />-valent star-like functions and weakly star-like functions in a disc (see [[#References|[2]]], [[#References|[4]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722043.png" />-locally star-like functions (see [[#References|[1]]]), and functions which are star-like in the direction of the real axis (see [[#References|[3]]]). For star-like functions of several complex variables, see [[#References|[5]]].
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Attention has also been given to univalent star-like functions in an annulus (see [[#References|[1]]]), $p$-valent star-like functions and weakly star-like functions in a disc (see [[#References|[2]]], [[#References|[4]]]), $\epsilon$-locally star-like functions (see [[#References|[1]]]), and functions which are star-like in the direction of the real axis (see [[#References|[3]]]). For star-like functions of several complex variables, see [[#References|[5]]].
  
 
====References====
 
====References====

Latest revision as of 14:43, 30 July 2014

univalent star-like function

A function $w=f(z)$ which is regular and univalent in the disc $|z|<1$, $f(0)=0$, and maps $|z|<1$ onto a star-like domain with respect to $w=0$. A function $f(z)$, $f(z)\neq0$ in $0<|z|<1$, $f(0)=0$, $f'(0)\neq0$, regular in $|z|<1$, is star-like in this disc if and only if it satisfies the condition

$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>0,\quad|z|<1.$$

The family of star-like functions in $|z|<1$, normalized so that $f(0)=0$, $f'(0)=1$, forms the class $S^*$, which admits a parametric representation by Stieltjes integrals:

$$f(z)=z\exp\left[-2\int_{-\pi}^{\pi}\log(1-e^{-it}z)d\mu(t)\right],$$

where $\mu(t)$ is a non-decreasing function on $[-\pi,\pi]$, $\mu(\pi)-\mu(-\pi)=1$.

For the class $S^*$ the coefficient problem has been solved; sharp estimates have been found for $|f(z)|$, $|f'(z)|$, $\arg f(z)$, $\arg f'(z)$ (the argument of the function is the branch that vanishes at $z=0$). The extremal functions for these estimates are $f(z)=z/(1-e^{i\theta}z)^2$, where $\theta$ is real. The class $S^*$ of functions $f(z)$ is related to the class of functions $\phi(z)$, $\phi(0)=0$, $\phi'(0)=1$, that are regular and univalent in $|z|<1$ and map $|z|<1$ onto a convex domain, by the formula $z\phi'(z)=f(z)$.

A star-like function such that

$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>\alpha,\quad|z|,\alpha<1,$$

is called a star-like function of order $\alpha$ in $|z|<1$.

Attention has also been given to univalent star-like functions in an annulus (see [1]), $p$-valent star-like functions and weakly star-like functions in a disc (see [2], [4]), $\epsilon$-locally star-like functions (see [1]), and functions which are star-like in the direction of the real axis (see [3]). For star-like functions of several complex variables, see [5].

References

[1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2] J.A. Hummel, "Multivalent starlike functions" J. d'Anal. Math. , 18 (1967) pp. 133–160
[3] M.S. Robertson, "Analytic functions star-like in one direction" Amer. J. Math. , 58 : 3 (1936) pp. 465–472
[4] A.W. Goodman, "Open problems on univalent and mutivalent functions" Bull. Amer. Math. Soc. , 74 : 6 (1968) pp. 1035–1050
[5] I.I. Bavrin, "Classes of holomorphic functions of several complex variables and extremal problems for these classes of functions" , Moscow (1976) (In Russian)


Comments

References

[a1] A.W. Goodman, "Univalent functions" , 1 , Mariner (1983)
[a2] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a3] C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
How to Cite This Entry:
Star-like function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Star-like_function&oldid=14539
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article