Certain Convolution Operators for Meromorphic Functions

Abstract

Let Σ (p ∈ N) be the class of functions $f(z) = z^{-p} + \sum^{\infty}_{n = 1} a_{n-p} z^{n-p}$ analytic in 0 < |z| < 1. A convolution operator L _p (a, c) on Σp is introduced. This paper gives some sharp inequalities for f(z) ∈ Σ satisfying Re{(1 − α)z^pL _p (a, c) f(z) + αz^pL _p (a + 1, c) f(z)} > β, where α ≥ 0, β < 1, a > 0 and c ≠ 0, −1, −2,....