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''with respect to a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872101.png" />''
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''with respect to a fixed point $O$''
  
A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872102.png" /> in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872104.png" />, such that, for any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872105.png" />, the segment of the straight line from that point to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872106.png" /> lies entirely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872107.png" />.
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A domain $D$ in the complex space $\mathbf C^n$, $n\geq1$, such that, for any point of $D$, the segment of the straight line from that point to $O$ lies entirely in $D$.
  
A simply-connected open [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872108.png" /> over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s0872109.png" />-plane is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721011.png" />-sheeted star-like domain with respect to a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721012.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721013.png" /> is a natural number) if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721014.png" /> points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721015.png" /> above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721016.png" /> (counting multiplicities) and if, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721017.png" />, there is a path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721018.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721019.png" /> to one of the points above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721020.png" /> such that the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721021.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721022.png" />-plane is the straight-line segment joining the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721024.png" />.
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A simply-connected open [[Riemann surface|Riemann surface]] $D$ over the $w$-plane is called a $p$-sheeted star-like domain with respect to a fixed point $a\in D$ (where $p$ is a natural number) if there exist $p$ points of $D$ above $w=a$ (counting multiplicities) and if, for any point $Q\in D$, there is a path $\Gamma\subset D$ from $Q$ to one of the points above $w=a$ such that the projection of $\Gamma$ on the $w$-plane is the straight-line segment joining the projection of $Q$ to $w=a$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721025.png" /> be a doubly-connected domain in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721026.png" />-plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721028.png" /> be complementary continua, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721029.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721030.png" /> be a fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721031.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721033.png" /> be the boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721034.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721035.png" /> is said to be star-like with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721036.png" /> if either each of the simply-connected domains containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721037.png" /> and bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721039.png" /> is star-like, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721040.png" /> is the union of the straight-line segments issuing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721042.png" /> is star-like with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721043.png" />.
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Let $B$ be a doubly-connected domain in the $w$-plane, let $E_1$ and $E_2$ be complementary continua, $\infty\in E_2$, let $a$ be a fixed point of $E_1$, and let $\Gamma_1$ and $\Gamma_2$ be the boundary components of $B$. Then $B$ is said to be star-like with respect to $a$ if either each of the simply-connected domains containing $a$ and bounded by $\Gamma_1$ and $\Gamma_2$ is star-like, or $\Gamma_1$ is the union of the straight-line segments issuing from $a$ and $E_1\cup B$ is star-like with respect to $a$.
  
 
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====Comments====
 
====Comments====
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087210/s08721044.png" />, star-like domains are the images of the unit disc under star-like functions (cf. [[Star-like function|Star-like function]]).
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For $n=1$, star-like domains are the images of the unit disc under star-like functions (cf. [[Star-like function|Star-like function]]).

Latest revision as of 14:28, 30 July 2014

with respect to a fixed point $O$

A domain $D$ in the complex space $\mathbf C^n$, $n\geq1$, such that, for any point of $D$, the segment of the straight line from that point to $O$ lies entirely in $D$.

A simply-connected open Riemann surface $D$ over the $w$-plane is called a $p$-sheeted star-like domain with respect to a fixed point $a\in D$ (where $p$ is a natural number) if there exist $p$ points of $D$ above $w=a$ (counting multiplicities) and if, for any point $Q\in D$, there is a path $\Gamma\subset D$ from $Q$ to one of the points above $w=a$ such that the projection of $\Gamma$ on the $w$-plane is the straight-line segment joining the projection of $Q$ to $w=a$.

Let $B$ be a doubly-connected domain in the $w$-plane, let $E_1$ and $E_2$ be complementary continua, $\infty\in E_2$, let $a$ be a fixed point of $E_1$, and let $\Gamma_1$ and $\Gamma_2$ be the boundary components of $B$. Then $B$ is said to be star-like with respect to $a$ if either each of the simply-connected domains containing $a$ and bounded by $\Gamma_1$ and $\Gamma_2$ is star-like, or $\Gamma_1$ is the union of the straight-line segments issuing from $a$ and $E_1\cup B$ is star-like with respect to $a$.

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


Comments

For $n=1$, star-like domains are the images of the unit disc under star-like functions (cf. Star-like function).

How to Cite This Entry:
Star-like domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Star-like_domain&oldid=32566
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article