Convolution conditions for bounded \(\alpha\)-starlike functions of complex order

A Y Lashin

doi:10.17951/a.2017.71.1.65

Abstract

Let \(A\) be the class of analytic functions in the unit disc \(U\) of the complex plane \(\mathbb{C}\) with the normalization \(f(0)=f^{^{\prime }}(0)-1=0\). We introduce a subclass \(S_{M}^{\ast }(\alpha ,b)\) of \(A\), which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class \(S_{M}^{\ast }(n,\alpha ,b)\) (\(n\geq 0\)) related to \(S_{M}^{\ast }(\alpha ,b)\) is also considered under the same conditions. Among other things, we find convolution conditions for a function \(f\in A\) to belong to the class \(S_{M}^{\ast }(\alpha ,b)\). Several properties of the class \(S_{M}^{\ast }(n,\alpha ,b)\) are investigated.

Highlights

Let H denote the class of analytic functions in the unit disc U = {z ∈ C : |z| < 1}
First let us define the class SM∗ (α, b) which unifies the classes of bounded starlike and convex functions of complex order
We say that f ∈ A belongs to the class SM∗ (α, b) (b ∈ C∗, α of bounded α-starlike functions of complex order, if and only if f (z)f z (z)

Summary

Introduction

Be the class of bounded starlike functions f satisfying the condition zf (z) − M ≤ M (z ∈ U ) . Of bounded starlike functions of complex order, if and only if f (z) z zf (z) f (z) First let us define the class SM∗ (α, b) which unifies the classes of bounded starlike and convex functions of complex order.

Results

Conclusion