ON PROPERTIES OF CERTAIN ANALYTIC MULTIPLIER TRANSFORM OF COMPLEX ORDER

The focus of this paper is to investigate the subclasses S∗C(γ, µ, α, λ; b), TS∗C(γ, µ, α, λ; b) = T∩S∗C(γ, µ, α, λ; b) and obtain the coefficient bounds as well as establishing its relationship with certain existing results in the literature.


Introduction
Let A be the class of normalized analytic functions f in the open unit disc U = {z ∈ C : |z| < 1} with f (0) = f (0) = 0 and of the form f (z) = z + ∞ n=2 a n z n , a n ∈ C, (1.1) and S the class of all functions in A that are univalent in U . Also, the subclass of functions in A that are of the form a n z n , a n ≥ 0, (1.2) is denoted by T and the subclasses S * (α), C(γ) are given respectively by Moreover, the class T S * (γ) denoted by T ∩ S * (γ) which is the subclass of function f ∈ T such that f is starlike of order γ and respectively, T C(γ) is the class of function f ∈ T such that f is convex of order γ.
Furthermore, the class T S * C(γ, β) which is the subclass of function f ∈ T such that f belongs the class S * C(γ, β), was studied by Altintas et al. and other researchers. For details see [ 3,5,6 ].
Using the unification in (5), Nizami Mustafa [6] introduced and investigated the class S * C(γ, β; τ ) and T S * C(γ, β; τ ), 0 ≤ α < 1; β ∈ [0, 1]; τ ∈ C which he defined as follows A function f ∈ S given by (1.1) is said to belong to the class S * C(γ, β; τ ) if the following condition is satisfied Meanwhile, the author in [4] defined a linear transformation D m α,λ f by Motivated by the work of Mustafa in [6], we study the effect of the application of the linear operator D m α,λ f on the unification of the classes of the functions S * C(γ, β; τ ). Now, we define the class S * C(γ, α, λ; b) to be class of functions f ∈ S which satisfies the condition Also, we denote by D T the subclass of the class of functions in (7) which is of the form and denote by T S * C(γ, µ, α, λ; b) = T ∩ S * C(γ, µ, α, λ; b) which is the class of functions f in (1.9) such that f belong to the class S * C(γ, µ, α, λ; b) = T ∩ S * C(γ, µ, α, λ; b).
In this paper, we investigate the subclasses S * C(γ, µ, α, λ; b) and 2. Coeffiecient bounds for the classes S * C λ α (γ, µ; b) and T S * C λ α (γ, µ; b) Theorem 2.1. Let f be as defined in (1.1). Then the function D m α,λ f belongs to the class S * C(γ, µ, α, λ; b), The result is sharp for the function It suffices to show that: Simple computation in (2.1), using (1.7), we have: Which implies that  (1) and the function D m α,λ f belongs to the class S * C(γ, µ, α, λ; b), The result is sharp for the function The result is sharp for the function The result is sharp for the function z n , n ≥ 2 Corollary 2.5. Let f be as defined in (1.1). Then the function D m α,λ f belongs to the class S * C(γ, µ, 1, 0, 1; b), The result is sharp for the function This result agrees with the Theorem 2.1 in [6].
Corollary 2.6. Let f be as defined in (1.1). Then the function D m α,λ f belongs to the class S * C(γ, 0, 1, λ, 0; 1), The result is sharp for the function This result agrees with the Corollary 2.1 in [6].
Corollary 2.7. Let f be as defined in (1.1). Then the function D m α,λ f belongs to the class S * C(γ, µ, 1, λ, 0; 1), The result is sharp for the function This result agrees with the Corollary 2.2 in [6]. Proof. We shall prove only the necessity part of the Theorem as the sufficiency proof is similar to the proof of Theorem 1.