Some subclasses of analytic functions of complex order

Nizami Mustafa

doi:10.3906/mat-1805-82

Abstract

Some subclasses of analytic functions of complex order

Highlights

Introduction and preliminariesLet A be the class of analytic functions f (z) in the open unit disk U = {z ∈ C : |z| < 1}, normalized by f (0) = 0 = f ′(0) − 1 of the form∑ ∞ f (z) = z + a2z2 + a3z3 + · · · + anzn + · · · = z + anzn, an ∈ C, n=2 (1.1)and S denote the class of all functions in A that are univalent in U
We will denote by S∗(α) and C(α) the subclasses of S that are, respectively, starlike and convex functions of order α (α ∈ [0, 1))
Coefficient bounds for the classes S∗C(α, β; τ ) and T S∗C(α, β; τ ) we will examine some inclusion results of the subclasses S∗C(α, β; τ ) and T S∗C(α, β; τ ) of analytic functions in the open unit disk

Summary

Introduction and preliminaries

Definition 1.1 A function f ∈ S given by (1.1) is said to be in the class S∗C(α, β; τ ) , α ∈ [0, 1) , β ∈ [0, 1] , τ ∈ C∗ = C − {0} if the following condition is satisfied:. Two new subclasses, S∗C(α, β; τ ) and T S∗C(α, β; τ ) , of the analytic functions in the open unit disk are introduced. 2. Coefficient bounds for the classes S∗C(α, β; τ ) and T S∗C(α, β; τ ) we will examine some inclusion results of the subclasses S∗C(α, β; τ ) and T S∗C(α, β; τ ) of analytic functions in the open unit disk.

It suffices to show that

The expression

Proof write