Certain Properties of a Generalized Class of Analytic Functions Involving Some Convolution Operator

Faroze Ahmad Malik,Chitaranjan Sharma,Nusrat Ahmed Dar

doi:10.34198/ejms.7121.4976

Faroze Ahmad Malik, Chitaranjan Sharma + Show 1 more

Open Access

https://doi.org/10.34198/ejms.7121.4976

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Abstract

We use the concept of convolution to introduce and study the properties of a unified family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$, $(0\leq\gamma\leq1,\,k\geq0)$, consisting of uniformly $k$-starlike and $k$-convex functions of complex order $b\in\mathbb{C}\setminus\{0\}$ and type $\alpha\in[0,1)$. The family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$ is a generalization of several other families of analytic functions available in literature. Apart from discussing the coefficient bounds, sharp radii estimates, extreme points and the subordination theorem for this family, we settle down the Silverman's conjecture for integral means inequality. Moreover, invariance of this family under certain well-known integral operators is also established in this paper. Some previously known results are obtained as special cases.

Highlights

Let H := H(D) be the collection of all functions f (z) that are analytic in the open unit disc D := {z ∈ C : |z| < 1}, and let A denote the class of normalized analytic
Bukhari et al [11] extended the idea of Aouf et al [4] to define a new class UM(g, γ, b, k) of analytic functions involving complex order as follows: Definition 1.2
Motivated by the above mentioned works, in this paper, we extend the class UM(g, γ, b, k) to the function class UMγ(g, b, k, α) which consists of analytic functions of complex order b and type α

Summary

Introduction

Let H := H(D) be the collection of all functions f (z) that are analytic in the open unit disc D := {z ∈ C : |z| < 1}, and let A denote the class of normalized analytic. To include the uniformly k-convex functions of order α, Aouf et al [4], in 2010, extended this class to S(g, γ, α, k) consisting of f (z) satisfying z(f ∗ g) (z) + γz2(f ∗ g) (z) −α (1 − γ)(f ∗ g)(z) + γz(f ∗ g) (z) z(f ∗ g) (z) + γz2(f ∗ g) (z). Bukhari et al [11] extended the idea of Aouf et al [4] to define a new class UM(g, γ, b, k) of analytic functions involving complex order as follows: Definition 1.2. Motivated by the above cited works on functions with negative coefficients, in this paper, we study various characteristic properties of the function class T U Mγ(g, b, k, α given by. F ∈ A will mean f (z) given in (1.1) and f ∈ T will mean f (z) given in (1.5), unless stated otherwise

Coefficient Estimates and Radii Problems

Integral Means Property

Subordination Theorem

Concluding Remarks