Majorization for certain classes of meromorphic functions defined by integral operator

S P Goyal,Pranay Goswami

doi:10.2478/v10062-012-0013-1

Abstract

Here we investigate a majorization problem involving starlike meromorphic functions of complex order belonging to a certain subclass of meromorphic univalent functions defined by an integral operator introduced recently by Lashin.

Highlights

Introduction and preliminariesLet f (z) and g(z) be analytic in the open unit disk (1.1)= {z ∈ C and |z| < 1}.For analytic functions f (z) and g(z) in ∆, we say that f (z) is majorized by g(z) in ∆ and write (1.2)f (z) g(z) (z ∈ ∆), if there exists a function φ(z), analytic in ∆ such that |φ(z)| ≤ 1, and (1.3)f (z) = φ(z)g(z) (z ∈ ∆).Let Σ denote the class of meromorphic functions of the form (1.4) f (z) = 1 + z ∞ ak z k, k=1
A function f (z) ∈ Σ is said to be in the class Sβα,j(γ) of meromorphic functions of complex order γ = 0 in ∆ if and only if
A majorization problem for the normalized classes of starlike functions has been investigated by Altinas et al [1] and MacGregor [9]

Summary

Introduction

For analytic functions f (z) and g(z) in ∆, we say that f (z) is majorized by g(z) in ∆ (see [9]) and write F (z) g(z) (z ∈ ∆), if there exists a function φ(z), analytic in ∆ such that |φ(z)| ≤ 1, and Let Σ denote the class of meromorphic functions of the form Meromorphic univalent functions, majorization property, starlike functions, integral operators.

Results

Conclusion