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Star-like domain

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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://www.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