Certain Subclass of Analytic Functions Deﬁned by Wanas Operator

In present article, we introduce and study a certain family of analytic functions deﬁned by Wanas operator in the open unit disk. We establish some important geometric properties for this family. Further we point out certain special cases for our results. ,


Introduction and Definitions
Let A denote the class of function f(z) which are normalized analytic in the open unit disk U = {z ∈ C : |z| < 1}. Let L(U) be the space of analytic functions in the unit disc U. Let A e = f ∈ L(U) : f(z) = z + σ e+1 z e+1 + σ e+2 z e+2 + · · · (1.1) with A 1 = A, z ∈ U and L[σ, e] := ϕ ∈ W(U) : ϕ(z) = σ + σ e z e + σ e+1 z e+1 + · · · (1.2) for σ ∈ C and e ∈ N . We denote by S subclass of functions which are analytic, univalent in U and has the normalization which implies that A function f(z) ∈ S is said to be a starlike, convex and turning bounded functions of order δ which are denoted by S * (δ), K(δ), R(δ) ⊂ S, if the following conditions are satisfied: f (z) + 1 > δ and (f (z)) > δ, where 0 δ < 1. Next we recall the definition of subordination. For two functions h 1 , h 2 ∈ U, we say that h 1 is subordinated to h 2 and symbolically written as h 1 ≺ h 2 if there exists an analytic function w with the property |w (z)| |z| such that h 1 (z) = h 2 (w (z)) for z ∈ D. Further, if h 2 ∈ S, then the condition becomes Wanas [16] in 2019 introduced the following operator, which can also be called (Wanas operator) W α,σ β,n : U −→ U defined by Special cases of this operator can be found in [1,2,3,6,9,10,13,14,15]. For more details see [17].
It is readily confirmed from (1.4) that

Main Result
is a new comprehensive class of holomorphic functions which includes numerous new classes of holomorphic univalent functions as well as some very well-known ones. In place of "equivalence" we are going to take "contained in" as it was discussed in [4], also see [11,12] for more details.   introduced by Cȃtaş and Lupas [5].
Varying the parameters of the above theorem gives the following corollaries.
Which implies that f(z) is convex of order 1/2.
Which implies that f(z) is a starlike function.