Univalence Conditions of Two New Integral Operators on P-Valent Functions

Abstract For analytic functions in the open unit diskU, we define two new general integral operators. The main object of the this paper is to study these two new integral operators and to determine some sufficient conditions for general p-valent integral operator to be p-th power of a univalent functions.


Introduction
Let A p be the class of functions of the form: which are analytic and p-valent in the unit disk U = {z : |z| < 1}.
A function f ∈ A p is called p-valent starlike of order γ if f (z) satisfies Re (1.2) 0 ≤ γ < p, p ∈ N * .We denote by S * p (γ) the class of all such functions.A function f ∈ A p is in the class S * p (γ), 0 ≤ γ < p, p ∈ N * if it satisfies the condition , we denote the class of all p-valent convex function of order γ.From (1.2) and (1.4), C ¸aǧlar showed us that: Now we define two new general p-valent integral operators.
The first new p-valent integral operator has the following form: where the functions f i ∈ A p , i = 1, 2, ..., n, and the paramentes β and α i , α i 0, Re(β) > 0 for all i = 1, 2, ..., n are complex numbers such that the integral operators in (1.7).
The second new p-valent integral operator has the form where the functions f i , g i ∈ A p , i = 1, 2, ..., n, and the paramentes β and α i , α i 0, Re(β) > 0 for all i = 1, 2, ..., n are complex numbers such that the integral operators in (1.8).Hallenbeck and Livingston defined p-subordination chains method and they obtained some results for f ∈ A p to be the p-th power of a univalent functions in U.
Theorem 1.1.Let f ∈ A p and α complex number such that Reα > 0 and is true for all z ∈ U, then the integral operator is the p-th power of a univalent function in U where the principal branch is considered.

Main Results
Firstly, we obtain sufficient conditions for p-valent integral operator defined by (1.7), to be the p-th power of a univalent function in U.
then the integral operator J p α i ,β defined by (1.7) is the p-th power of a univalent function in U.
Proof.We define the function 3) logarithmically and multiplying by z, we get (2.4) Thus, we obtain From the hypothesis of the Theorem 2.1, we get: (2.6) Applying Theorem 1.1, we get the integral operator J p α i ,β defined by (1.7) is the p-th power of a univalent function in U.
then the integral operator is the p-th power of a univalent function in U.
Corollary 2.2.Let the function f ∈ S * p (γ), 0 ≤ γ < p, p ∈ N * , then the integral operator is the p-th power of a univalent function in U.
In the next theorem, we derive another sufficient condition for p-valent integral operator defined by (1.8), to be the p-th power of a univalent function in U.
Theorem 2.2.Let the functions f i ∈ S * p (γ i ) and then the integral operator I p α i ,β defined by (1.8) is the p-th power of a univalent function in U.
Proof.We define (2.11) It is easy to see that (2.12) Differentiating (2.12) logarithmically and multiplying by z, we obtain (2.13) Thus, we get From the hypothesis of the Theorem 2.2, we get (2.15) Applying Theorem 1.1, we get the integral operator I p α i ,β defined by (1.8) is the p-th power of a univalent function in U.