Topological Properties for Harmonic -Uniformly Convex Functions of Order Associated with Wanas Differential Operator

The purpose of the present paper is to establish some topological properties for a certain family of harmonic -uniformly convex functions of order associated with Wanas differential operator defined in the open unit disk .


Introduction
A continuous function = + is a complex valued harmonic function in a complex domain ℂ, if both and are real harmonic in ℂ. In any simply connected domain ⊂ ℂ, we can write = ℎ + , where ℎ and are analytic in . We call ℎ the analytic part and the co-analytic part of . A necessary and sufficient condition for to be locally univalent and sense-preserving in is that |ℎ′( )| > | ′( )| in (see Clunie and Sheil-Small [6]).
Each ∈ ℋ can be expressed as Also note that ℋ reduces to the family , of analytic functions in if co-analytic part of is identically zero.
We consider the usual topology on ℋ defined by a metric in which a sequence ' in ℋ converges to if and only if it converges to uniformly on each compact subset of . It follows from the theorems of Weierstrass and Montel that this topological space is complete.
We denote by Eℳ the set of all extreme points of ℳ. It is clear that Eℳ ⊂ ℳ.
Moreover, we define the closed convex hull of ℳ as the intersection of all closed convex subsets of ℋ (with respect to the topology of locally uniform convergence) that contain ℳ. We denote the closed convex hull of ℳ by HI JJJ ℳ.
Also denote by j ℋ the sub-family of ℋ containing of all functions = ℎ + , where ℎ and are given by It is easily verified that if ∈ j ℋ , we also have

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We now recall the following lemmas that will be used to prove our main results.

A Set of Main Results
In the first theorem, we determine the sufficient condition for = ℎ + to be in the family V ℋ ( , , N, O, ], R).
Theorem 2.1. Let = ℎ + with ℎ and are given by or equivalently We only need to prove that The next theorem shows that condition (2.1) is also the sufficient condition for functions ∈ j ℋ to be in the family Vj ℋ ( , , N, O, ], R).  is continuous and convex on ℋ.
By making use of Theorem 2.4 and Lemma 1.1, we obtain the following corollaries:  The function ℎ * is the function defined by (2.7).

Conclusion
The results we obtained in this paper which may be considered as a useful tool for those who are interested in the above-mentioned topics for further research. It may also be used to find prospective applications in some areas of mathematics and physics.