On certain general integral operators of analytic functions

In this paper, we obtain new sufficient conditions for the operators Fα1,α2,...,αn,β(z) and Gα1,α2,...,αn,β(z) to be univalent in the open unit disc U , where the functions f1, f2, . . . , fn belong to the classes S(a, b) and K(a, b). The order of convexity for the operators Fα1,α2,...,αn,β(z) and Gα1,α2,...,αn,β(z) is also determined. Furthermore, and for β = 1, we obtain sufficient conditions for the operators Fn(z) and Gn(z) to be in the class K(a, b). Several corollaries and consequences of the main results are also considered.


Introduction and definitions.
Let A denote the class of functions of the form a n z n which are analytic in the open unit disc U = {z : |z| < 1}.Further, by S we shall denote the class of all functions in A which are univalent in U. A function f (z) ∈ A is said to be starlike of order γ (0 ≤ γ < 1) if it satisfies (1.1) Re zf (z) Also, we say that a function f (z) ∈ A is said to be convex of order γ (0 ≤ γ < 1) if it satisfies We denote by S (γ) and K(γ) , respectively, the usual classes of starlike and convex functions of order γ (0 ≤ γ < 1) in U.
A function f ∈ A is said to be in the class S (a, b) if and a function f ∈ A is said to be in the class K(a, b) if From (1.3) and (1.4), we have and The class S (a, b) was introduced by Jakubowski [12].
Further, applying the Briot-Bouquet differential subordination [9], we can easily see that K(a, b) ⊂ S (a, b).
Several authors (e.g., see [4,5,6,8,10,11,15,16]), obtained many sufficient conditions for the univalency of the integral operators , where the functions f 1 , f 2 , . . ., f n belong to the class A and the parameters α 1 , α 2 , . . ., α n , and β are complex numbers such that the integrals in (1.5) and (1.6) exist.Here and throughout in the sequel every many-valued function is taken with the principal branch.
In the proofs of our main results we need the following univalence criteria.The first result, i.e.Lemma 1.1 is a generalization of the wellknown univalence criterion of Becker [2] (which in fact corresponds to the case β = δ = 1), while the second, i.e.Lemma 1.2 is a generalization of Ahlfors' and Becker's univalence criterion [1,3] (which corresponds to the case β = 1).

Lemma 1.1 ([13]
).Let δ be a complex number with Re(δ for all z ∈ U, then, for any complex number β with Re(β) ≥ Re(δ), the integral operator is in the class S.

Lemma 1.2 ([14]
).Let β be a complex number with Re(β) > 0 and c be a complex number with |c| ≤ for all z ∈ U, then the integral operator is in the class S.
is analytic and univalent in U.

Proof.
Defining we observe that h(0) = h (0) − 1 = 0. On the other hand, it is easy to see that Differentiating both sides of (3.1) logarithmically, we obtain Thus, we have is analytic and univalent in U.

Order of convexity. Now, we prove
Then the integral operator F n (z) defined by (1.7) is in the class Proof.From (1.7), it follows that (4.1) Differentiating both sides of (4.1) logarithmically, we obtain Then the integral operator G n (z) defined by (1.8) is in the class Proof.From (1.8), we have