Certain subclasses of starlike functions of complex order involving the Hurwitz—Lerch Zeta function

G Murugusundaramoorthy,K Uma

doi:10.2478/v10062-010-0015-9

Abstract

Making use of the Hurwitz-Lerch Zeta function, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients of complex order denoted by \(TS^\mu_b(\alpha, \beta, \gamma)\) and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class \(TS^\mu_b(\alpha, \beta, \gamma)\). In particular, we obtain integral means inequalities for the function \(f(z)\) belongs to the class \(TS^\mu_b(\alpha, \beta, \gamma)\) in the unit disc.

Highlights

Let A denote the class of functions of the form ∞ (1.1)f (z) = z + anzn n=2 which are analytic and univalent in the open disc U = {z : z ∈ C, |z| < 1}. denote by T a subclass of A consisting of functions of the form (1.2)f (z) = z − anzn; an ≥ 0, z ∈ U, n=2 introduced and studied by Silverman [25]
Motivated by the study on uniformly convex and uniformly starlike functions and making use of the operator Jbμ, we introduce a new subclass of analytic functions with negative coefficients and discuss some usual properties of the geometric function theory of this generalized function class
The main object of this paper is to study some usual properties of the geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness and convexity for the class T Sbμ(α, β, γ)

Summary

Introduction

Let A denote the class of functions of the form Certain subclasses of starlike functions of complex order... Motivated by the study on uniformly convex and uniformly starlike functions (see [9, 10, 12, 13, 14, 15, 22, 23]) and making use of the operator Jbμ, we introduce a new subclass of analytic functions with negative coefficients and discuss some usual properties of the geometric function theory of this generalized function class.

Results

Conclusion