On certain classes of p-valent functions. (English) Zbl 0667.30013

Owa, Shigeyoshi

Čas. Pěstování Mat. 113, No. 4, 342-350 (1988).

Let \(D^{n+p-1}f(z)\) be defined to equal to \(\{\) z/(1-z)\(\}\) *f(z) where * denotes the Hadamard product. The author defines the classes \(T^*(n+p-1,\alpha)\) to consist of functions of the form \[ f(z)=z^ p- a_{p+1}z^{p+1}-a_{p+2}z^{p+2}-...\quad, \] analytic in the unit disc, such that all the \(a_ k\) are non-negative real numbers, and such that \(Re\{D^{n+p}f(z)/D^{n+p-1}f(z)\}>\alpha\) for all z in the unit disc. Several distortion theorems and closure theorems are proved.

Reviewer: J.A.Hummel

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

p-valent functions; Hadamard product; distortion theorems; closure theorems