Starlikeness Condition for a New Differential-Integral Operator

: A new differential-integral operator of the form I n f ( z ) = ( 1 − λ ) S n f ( z ) + λ L n f ( z ) , z ∈ U , f ∈ A , 0 ≤ λ ≤ 1, n ∈ N is introduced in this paper, where S n is the S˘al˘agean differential operator and L n is the Alexander integral operator. Using this operator, a new integral operator is deﬁned as: F ( z ) = (cid:20) , where I n f ( z ) is the differential-integral operator given above. Using a differential subordination, we prove that the integral operator F ( z ) is starlike.


Introduction and Preliminaries
The introduction and study of operators has been a topic that emerged at the very beginning of the theory of functions of a complex variable. The first operators were introduced during the first years of the twentieth century by mathematicians like J.W. Alexander, R. Libera, S. Bernardi, P. T. Mocanu and many more. The Alexander integral operator is such an example, defined by J. W. Alexander in 1915 [1]. This paper is cited in nearly 500 papers. The use of operators has facilitated the introduction of special classes of univalent functions and studying properties of the functions in those classes, such as convexity, starlikeness, coefficient estimates, and distortion properties. The Sȃlȃgean differential operator was introduced in 1983 [2] and is cited by over 1300 papers. It has been used in obtaining new classes of functions and proving many interesting results related to them. The operator we introduce in this paper gives a new perspective in the theory related to operators by combining the integral Alexander operator and the differential Sȃlȃgean operator. The results were obtained also using the means of the theory of differential subordinations introduced by Professors Miller and Mocanu in two papers in 1978 and 1980 and condensed in the monograph published by them in 2000 [3]. This theory has remarkable applications allowing easier proofs of already known results and facilitating the emergence of new ones. The idea of combining integral and differential operators is illustrated in the very recent paper [4] where a differential-integral operator was defined and using the method of the subordination chains, differential subordinations in their special Briot-Bouquet form were studied obtaining their best dominant and, as a consequence, criteria containing sufficient conditions for univalence were formulated. Similar work containing subordination results related to a class of univalent functions obtained by the use of an operator introduced by using a differential operator and an integral one can be seen in [5].
We use the well-known notations: • H(U) is the class of functions analytic in the unit disc U = {z ∈ C : |z| < 1}, The definitions of subordination, solution of the differential subordination and best dominant of the solutions of the differential subordination are recalled next as they can be found in the monograph published by Professors Miller and Mocanu in 2000 [3], which gives the core of the theory of differential subordination: If f and g are analytic in U, then we say that f is subordinate to g, Let ψ : C 3 × U → C and h be a univalent function in U. If p is a analytic function in U which satisfies the following (second-order) differential subordination: then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination or more simply a dominant, if p ≺ q, for all p satisfying the differential subordination. A dominant q that satisfies q ≺ q for every dominant q is said to be the best dominant.
A well-known lemma from the theory of differential subordinations that is used in proving the new results is shown as follows: Lemma 1. [3] Let g be univalent in U and let θ and φ be analytic in a domain D containing g(U), with φ(w) = 0, when w ∈ g(U). Set and suppose that (i) Q is starlike and then p(z) ≺ q(z), and q is the best dominant.
In order to define the new differential-integral operator, we need the following definitions: Definition 1. [2] For f ∈ A, n ∈ N = N * ∪ {0}, let S n be the differential operator given by S n : Definition 2.
[6] For f ∈ A, n ∈ N = N * ∪ {0}, let L n be the integral operator given by L n : A → A with (b) For f ∈ A, f (z) = z + ∞ ∑ j=2 a j z j , we obtain:

Main Results
Using Definition 1 and Definition 2, we introduce a new operator, as follows: Denote by I n the differential-integral operator I n : A → A given by where S n is Sȃlȃgean differential operator, and L n is Alexander integral operator.

Remark 3. (a)
For λ = 0, I n f (z) = S n f (z), the differential-integral operator is equivalent to Sȃlȃgean differential operator.
(b) For λ = 1, I n f (z) = L n f (z), the differential-integral operator becomes Alexander integral operator.
(c) For f ∈ A, f (z) = z + ∞ ∑ j=2 a j z j we obtain Using the differential-integral operator introduced in Definition 3, we define a new integral operator, which can be seen as generalization of some well-known integral operators.

Remark 4. (a) For n
which is the Bernardi integral operator [7].
which is the Libera integral operator [8].
(e) For n = 0, we have [9] where the authors have proved that F ∈ S * .
Using a differential subordination, we prove that the operator given by Equation (5) is starlike.
If f ∈ A and 1 β · z(I n f (z)) I n f (z) then is starlike, i.e., F ∈ S * , where I n f is given by Equation (3).
If we let θ : D ⊂ C → C and φ : D ⊂ C → C be analytic, and Next we show that conditions in Lemma 1 are satisfied. We prove that the function Q is starlike. Differentiating Equation (15), we have We take Re We have shown that Re zQ (z) Q(z) > 0, z ∈ U, i.e., Q ∈ S * , hence (i) from Lemma 1 is satisfied.