On meromorphic starlike and convex functions. (English) Zbl 1013.30005

Nunokawa, M.; Ahuja, O. P.

Indian J. Pure Appl. Math. 32, No. 7, 1027-1032 (2001).

Let \(\Sigma\) denote the class of functions \(F\) of the form \(F(z)=\frac{1}{z}+\sum_{n=0}^{\infty}a_{n}z^{n}\), which is analytic and univalent in \(D=\{z:0<|z|<1\}\). A function \(F\in\Sigma\) denoted by \(MS^{*}(\alpha)\) is called meromorphic starlike of order \(\alpha(\alpha<1)\), if \(F(z)\neq 0\) in \(D\) and \(-\text{Re}\dfrac{zF'(z)}{F(z)}>\alpha\), \(z\in E\), \(E=\{z:|z|<1\}\). Similarly, a function \(F\in\Sigma\) denoted by \(MC(\alpha)\) is called meromorphic convex of order \(\alpha<1\) if \(F(z)\neq 0\) in \(D\) and \(-(1+\text{Re}\dfrac{zF''(z)}{F'(z)})>\alpha\), \(z\in E\). The author shows that the aforementioned functions do not hold the same relationship to the usual starlike univalent functions and convex univalent functions.

Reviewer: Maslina Darus (Bangi Selangor)

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C55	General theory of univalent and multivalent functions of one complex variable

Keywords:

meromorphics functions; starlike and convex functions; conformal mapping