Study on Integral Operators by Using Komato Operator on a New Class of Univalent Functions

and we study linear combination and derive some interesting properties for the class Tc(α, β). Also, we study on some integral operators on Tc(α, β).


Introduction
Let A denote the class of functions analytic in the unit disk U = {z ∈ C : |z| < 1} and let T denotes the subclass of A consisting univalent functions of the from which are analytic in the unit disk U.

Main Results
We need the following elementary lemmas.
The proof of the following result, which is given in (Najafzadeh, 2009, p81-89), needs some corrections and is given for the convenience of the reader.
and by using (4) we may write . By Lemma 2.1 and letting w = and so by the mean value theorem we have and the proof is complete.
Theorem 2.6.The function F(z) defined upon belongs to T δ c (α, β).Proof.We have This shows that F(z) ∈ T δ c (α, β) and so the proof is completed.
Therefore it is enough )( c+1 c+k ) 2 and this gives the result.

Study on some of Integral operators on T
Next we investigate some of properties of the function F μ (z) in the class T δ c (α, β).
So the proof is completed.
Therefore it is enough, by Theorem 2.3 and Corollary 2.4, letting Proof.For r 2 (α, β, η), we must show that zF μ (z) Therefore it is enough, by Theorem 2.3 and Corollary 2.4, that This shows that F μ (z) is close-to-convex of order μ.