Meromorphic multivalent functions with positive coefficients. I. (English) Zbl 0705.30019

Mogra, M. L.

Math. Jap. 35, No. 1, 1-11 (1990).

Let \(\Sigma\) (p;A,B), \(p\in {\mathbb{N}}\), \(0\leq B\leq 1\), \(-B\leq A<B\), denote the class of functions of the form \[ f(z)=z^{-p}+a_ pz^ p+a_{p+1}z^{p+1}+... \] such that \[ \frac{zf'(z)}{f(z)}=- p\frac{1+A\omega (z)}{1+B\omega (z)},\text{ where } | \omega (z)| \leq | z| <1, \] and let \(\Sigma^*(p;A,B)\) denote the subclasses of \(\Sigma\) (p;A,B) of the functions with positive coefficients \((a_ k\geq 0\), \(k=p,p+1,...).\)
In this paper it is proved that for \(f\in \Sigma^*(p;A,B)\) \[ | f(z)| \leq | z|^ p+\frac{B-A}{2+B+A}| z|^ p. \] Some coefficients inequalities and radius of convexity are derived. The extreme points and some properties of convolution for the class \(\Sigma^*(p;A,B)\) are also investigated.

Reviewer: J.Stankiewicz

MSC:

30C50	Coefficient problems for univalent and multivalent functions of one complex variable
30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

extreme points; convolution