A New Differential Operator of Analytic Functions Involving Binomial Series

then p is a solution of the differential superordination (1.2). (If f is subordinate to F , then F is superordinate to f .) An analytic function q is called a subordinant if q ≺ p for all p satisfying (1.2). A univalent subordinant q̃ that satisfies q ≺ q̃ for all subordinants q of (1.2) is said to be the best subordinant. Miller and Mocanu [7] obtained conditions on h, q and φ for which the following implication holds: h(z) ≺ φ(p(z), zp(z), zp(z); z) ⇒ q(z) ≺ p(z).


Introduction and Definitions
Let H be the class of functions analytic in U := {z : |z| < 1} and H(a, n) be the subclass of H consisting of functions of the form f (z) = a + a n z n + a n+1 z n+1 + . ... Let A be the subclass of H consisting of functions of the form (1.1) Let p, h ∈ H and let φ(r, s, t; z) : C 3 ×U → C. If p and φ(p(z), zp ′ (z), z 2 p ′′ (z); z) are univalent and if p satisfies the second order superordination h(z) ≺ φ(p(z), zp ′ (z), z 2 p ′′ (z); z), then p is a solution of the differential superordination (1.2). (If f is subordinate to F , then F is superordinate to f .)An analytic function q is called a subordinant if q ≺ p for all p satisfying (1.2).A univalent subordinant q that satisfies q ≺ q for all subordinants q of (1.2) is said to be the best subordinant.Miller and Mocanu [7] obtained conditions on h, q and φ for which the following implication holds: Using the results of Miller and Mocanu [7], Bulboacȃ [4] considered certain classes of first order differential superordinations as well as superordination-preserving integral operators [3] (see also, [1,5,10]).Shanmugam et al. [9] obtained sufficient conditions for a normalized analytic functions f (z) to satisfy where q 1 and q 2 are given univalent functions in U with q 1 (0) = 1 and q 2 (0) = 1.
For a function f in A, and making use of the binomial series we now define the differential operator D ζ m,λ f (z) as follows: If f is given by (1.1), then from (1.5) and (1.6) we see that Using the relation (1.7), it is easily verified that where We observe that for m = 1, we obtain the differential operator D ζ 1,λ defined by Al-Oboudi [2] and for m = λ = 1,we get Sȃlȃgean differential operator D ζ [8].
The main object of the present paper is to apply a method based on the differential subordination in order to derive several subordination results involving the operator D ζ m,λ .Furthermore, we obtain the previous results of Srivastava and Lashin [11] as special cases of some of the results presented here.

Preliminaries
In order to prove our results, we shall require the following known definition and lemmas.Theorem 3.4h,p. 132] Let q(z) be univalent in the unit disk U and θ and φ be analytic in a domain D containing q(U) with φ(w) = 0 when then p(z) ≺ q(z) and q(z) is the best dominant.

Subordination for Analytic Functions
We begin by proving the following result.
Lemma 3.1.Let the functions p(z) and q(z) be analytic in U and suppose that q(z) = 0 (z ∈ U) is also univalent in U and that If q(z) satisfies and then p(z) ≺ q(z) (z ∈ U) and q is the best dominant.
Then, we observe that θ(ω) Also, by letting and we find from (3.1) and (3.2), Q(z) is starlike univalent in U and that Our result now follows by an application of Lemma 2.2.✷ We first prove the following subordination theorem involving the operator D ζ m,λ .Theorem 3.2.Let the function q(z) be analytic and univalent in U such that q(z) = 0 (z ∈ U).Suppose that zq ′ (z) q(z) is starlike univalent in U and the inequality (3.2) holds true.Let and q is the best dominant.
A New Differential Operator of Analytic Functions Involving Binomial ... 211 and e ǫAz is the best dominant.
For q(z) = b in Theorem 3.2, we get the following result obtained by Srivastava and Lashin [11].
Corollary 3.6.Let b be a non zero complex number.If f ∈ A, and

Superordination for Analytic Functions
Next, applying Lemma 2.3, we obtain the following two theorems.Theorem 4.1.Let q be analytic and convex univalent in U such that q(z) = 0 and zq ′ (z) q(z) is starlike univalent in U. Suppose also that 3) is univalent in U, then the following superordination: and q(z) is the best subordinant.

Definition 2 . 1 .
[7, Definition 2, p. 817]  Denote by Q, the set of all functions f (z) that are analytic and injective on U − E(f ), where