Coefficient inequalities for concave and meromorphically starlike univalent functions

B. Bhowmik¹ ; S. Ponnusamy¹

¹Department of Mathematics Indian Institute of Technology Madras Chennai 600 036, India

Annales Polonici Mathematici, Tome 93 (2008) no. 2, pp. 177-186

Résumé

Let $\mathbb D$ denote the open unit disk and $f:\mathbb D\rightarrow\overline{\mathbb C}$ be meromorphic and univalent in $\mathbb D$ with a simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, assume that $f$ has the expansion $$ f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|1-p, $$ and maps $\mathbb D$ onto a domain whose complement with respect to $\overline{\mathbb C}$ is a convex set (starlike set with respect to a point $w_0\in \mathbb C, w_0\neq 0$ resp.). We call such functions concave (meromorphically starlike resp.) univalent functions and denote this class by ${\rm Co}(p)$ $({\mit\Sigma}^{\rm s}(p, w_0)$ resp.). We prove some coefficient estimates for functions in these classes; the sharpness of these estimates is also established.

Détail

DOI : 10.4064/ap93-2-6

Keywords: mathbb denote unit disk mathbb rightarrow overline mathbb meromorphic univalent mathbb simple pole satisfying standard normalization assume has expansion sum infty z p quad z p p maps mathbb domain whose complement respect overline mathbb convex set starlike set respect point mathbb neq resp call functions concave meromorphically starlike resp univalent functions denote class mit sigma resp prove coefficient estimates functions these classes sharpness these estimates established