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Difference between revisions of "Limit of star-likeness"

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''exact radius of star-likeness, bound of star-likeness''
 
''exact radius of star-likeness, bound of star-likeness''
  
The least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588701.png" /> of the radii of discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588702.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588703.png" /> is some class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588704.png" /> that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588705.png" />, such that the functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588706.png" /> on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588707.png" /> map the discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588708.png" /> onto star-like domains (cf. [[Star-like domain|Star-like domain]]) about the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l0588709.png" />. Any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887010.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887011.png" /> is called a radius of star-likeness of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887012.png" />.
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The least upper bound $R_U$ of the radii of discs $|z|\leq r$, where $U$ is some class of functions $w=f(z)+\dots$ that are regular and univalent in $|z|<1$, such that the functions from $U$ on the disc $|z|<1$ map the discs $|z|\leq r$ onto star-like domains (cf. [[Star-like domain|Star-like domain]]) about the point $w=0$. Any number $r$ in the interval $0<r<R_U$ is called a radius of star-likeness of the class $U$.
  
The limit of star-likeness is usually found by using the following criterion of star-likeness: A disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887013.png" /> is mapped onto a star-like domain by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887014.png" /> if and only if on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887015.png" />,
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The limit of star-likeness is usually found by using the following criterion of star-likeness: A disc $|z|<r$ is mapped onto a star-like domain by $w=f(z)$ if and only if on $|z|=r$,
  
<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/l/l058/l058870/l05887016.png" /></td> </tr></table>
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$$\frac{\partial\arg f(z)}{\partial\phi}=\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]\geq0,\quad z=re^{i\phi},$$
  
 
or, equivalently,
 
or, equivalently,
  
<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/l/l058/l058870/l05887017.png" /></td> </tr></table>
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$$\left|\arg\frac{zf'(z)}{f(z)}\right|\leq\frac\pi2.$$
  
The limit of star-likeness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887018.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887019.png" /> of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887020.png" /> that are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887021.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058870/l05887022.png" />.
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The limit of star-likeness $R_S$ of the class $S$ of all functions $f(z)=z+\dots$ that are regular and univalent in the disc $|z|<1$ is equal to $\tanh(\pi/4)=0.65\dots$.
  
 
====References====
 
====References====

Revision as of 08:50, 31 October 2014

exact radius of star-likeness, bound of star-likeness

The least upper bound $R_U$ of the radii of discs $|z|\leq r$, where $U$ is some class of functions $w=f(z)+\dots$ that are regular and univalent in $|z|<1$, such that the functions from $U$ on the disc $|z|<1$ map the discs $|z|\leq r$ onto star-like domains (cf. Star-like domain) about the point $w=0$. Any number $r$ in the interval $0<r<R_U$ is called a radius of star-likeness of the class $U$.

The limit of star-likeness is usually found by using the following criterion of star-likeness: A disc $|z|<r$ is mapped onto a star-like domain by $w=f(z)$ if and only if on $|z|=r$,

$$\frac{\partial\arg f(z)}{\partial\phi}=\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]\geq0,\quad z=re^{i\phi},$$

or, equivalently,

$$\left|\arg\frac{zf'(z)}{f(z)}\right|\leq\frac\pi2.$$

The limit of star-likeness $R_S$ of the class $S$ of all functions $f(z)=z+\dots$ that are regular and univalent in the disc $|z|<1$ is equal to $\tanh(\pi/4)=0.65\dots$.

References

[1] 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
How to Cite This Entry:
Limit of star-likeness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_of_star-likeness&oldid=13077
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article