CERTAIN CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS

Abstract

Let

\[ f(z) =\frac{1}{z^p}+\sum_{n=1}^\infty \frac{a_{n-1}}{z^{p-n}} \]

be regular in the punctured disk $E =\{z: 0<|z|<1\}$ and

\[ D^{n+p-1}f(z)=\frac{1}{z^p(1-z)^{n+p}}*f(z) \]

where $*$ denotes the Hadamard product and $n$ is any integer greater than $- p$. For $- 1 \le B < A \le 1$, let $C_{n,p}(A, B)$ denote the class of functions $f(z)$ satisfying

\[-z^{p+1}(D^{n+p-1}f(z))'<p\frac{1+Az}{1+Bz}\]

This paper establishes the property $C_{n+1,p}(A,B) \subset C_{n,p}(A,B)$. Fur­ ther property preserving integral operators, coefficient inequalities and a closure theorem for these classes are obtained. Our results generalise some of the recent results of Ganigi and Uralegaddi [1].