Univalence properties of an integral operator

In this paper, we determine conditions on β,αi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta , \alpha _{i}$$\end{document} and gi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i}$$\end{document} such that the integral operator Gα1,α2,…,αn,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ G_{\alpha _1, \alpha _2,\ldots ,\alpha _n, \beta }}$$\end{document} is univalent on the exterior of the unit disk for two subclasses of analytical functions.


Introduction
Let O be the class of analytical functions g defined on the exterior of the unit disk W = {z ∈ C : 1 < |z| < ∞}.
Let be the subclass of O which contains the univalent functions of W. Let O 1 be the subclass of O which contains the meromorphic, normalized and injective functions g : W −→ C ∞ , that looks like [2]: Let V 2 be the subclass of V . Let V 2,μ be the subclass of V 2 which contains the functions of the form (1.1) and satisfies the condition: for μ > 1 and we note V 2,1 = V 2 .
Let p ∈ R, with 1 < p < 2, S(p) is the subclass of O with all the functions such that: Then we obtain the inequation: Let A be the class of analytic functions f(z) defined in the open unit disk U := {z ∈ C : |z| < 1} and normalized by the conditions: Let S be the subclass of A consisting of univalent functions in U, of the form: Let T be the univalent subclass of A which satisfies: Let T 2 be the subclass of T for which f (0) = 0. It is known that between the S class and the class there are the following links: Proposition 1.1 [2] (i) Let f ∈S and g(ς)=1/f(1/ς), ς ∈ W . Then g ∈ and g(ς) =0, ς ∈ W .
Let F α 1 ,α 2 ,...,α n ,β be the integral operator introduced by Daniel Breaz and Narayanasamy Seenivasagan [3]: and we take into account that f i (t) ∈ S. ( f i (t) ∈ T 2 which is a subclass of T, which is a subclass of A). Let be g i (t) = 1 We remember that O 1 is the subclass of O with: We may say that between T 2 and O 1 there is a bijection. We start from: And we apply the following transformations: We can form the integral operator: Pascu proved the following theorem: Theorem 1.1 [4,5] Let β ∈ C, Reβ ≥ γ > 0. If the function f ∈ A satisfies the condition: then the integral operator: If the function f ∈ T 2 satisfies the condition: then the integral operator: is in S.
Using Theorem 1.1 and Theorem 1.2, D. Breaz and N. Breaz obtained the following theorem: then the integral operator: is in S.

Main results
Applying the Theorem 1.1 for the transformation t → 1 t |() , we obtain: then the integral operator: is in .
Proof Let be the function: We notice that: We know that: Applying the relation (2.9) in the relation (2.8) we get: We know that: Using the results from the Eqs. (2.10) and (2.11) we get: Given that Reβ ≥ γ > 0 result from Theorem 1.1. that: Meaning that the integral operator: is in . Theorem 2.5 Let m > 1, g i ∈ S( p) and: Then we obtain that the integral operator defined in (1.5) G α i ,β is in .
The proof of this theorem is very similar with the previous one.
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