On a Univalent Class Involving Differential Subordinations with Applications

Abstract: Problem statement: By means of the Hadamard product (or convolution), new class of analytic functions was formed. This class was motivated by many authors. Approach: By using the concept of the subordination and superordination, we define certain differential inequalities and first order differential subordinations. Results: As their applications, we obtain some sufficient conditions for univalence which generalize and refine some previous results. Sandwich theorem is also obtained. Conclusion: Therefore, we posed a new class of analytic functions which generalized some well known subclasses. This class involves the E( , ) Φ Ψ − family of functions.


INTRODUCTION
Let F be analytic in U, g analytic and univalent in U and f (0)= g (0) Then, by the symbol f (z) g(z) ≺ (f subordinate to g) in U, we shall mean f (U) g(U). ⊂ Let 2 : C C ϕ → and let h be univalent in U. If p is analytic in U and satisfies the differential subordination (p(z)), zp (z)) h(z) ′ ϕ ≺ 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, if p q.

≺
If P and (p(z)), zp (z)) ′ ϕ are univalent in U and satisfy the differential superordination h(z) (p(z)), zp (z)) ′ ϕ ≺ then P is called a solution of the differential superordination. An analytic function q is called subordinant of the solution of the differential superordination if q p. ≺ Juneja defined the family E( , ), Φ Ψ so that: This type of class was motivated by many authors namely (Lewandowski et al.,1976;Kumar et al., 1995;Kwon , 2007;Ravichandran et al., 2002;Obradovi´c and Joshi, 1998;Joshi et al., 1998;Singh and Gupta, 1996;Xu and Yang, 2005). Note that this family was then extended and studied in the work due to Ibrahim and Darus.

MATERIALS AND METHODS
In the present study, we consider a new class H( , , , (z); (z)) α λ δ Φ Ψ as follows Eq. 1: and F is the conformal mapping of the unit disk U with F (0)= 1.
Remark 1: As special cases of the class H( , , , (z); (z)) α λ δ Φ Ψ are the following well known classes: Lewandowski et al., 1976) ( Xu and Yang, 2005). Also this class reduces to the classes of starlike functions, convex functions and close-to-convex functions for various Φ and ψ.
In order to obtain our results, we need the following lemmas. (2000). Let q (z) be univalent in the unit disk U and θ and ϕ be analytic in a domain D containing q (U) with (w) 0 ϕ ≠ when w q(U). ∈ Set:

Lemma 1: Miler and Mocanu
Definition 1: (Miller and Mocanu, 2003) Denote by Q the set of all functions f(z) that are analytic and injective on and are such that Lemma 2: Bulboaca (2002). Let q(z) be convex univalent in the unit disk U and ϑ and ϕ be analytic in a domain D containing q(U) Suppose that :

RESULTS AND DISCUSSION
In this section, we prove a subordination theorem by using Lemma 1 and as applications of this result, we find the sufficient conditions for f∈A to be univalent.
Theorem 1: Let q,q(z)≠ 0 be a univalent function in U and g(z)≠ 0 be analytic in C such that for nonnegative real numbers μ and ν Eq. 2: If p(z) 0,z U ≠ ∈ satisfies the differential subordination: then p q ≺ and q is the best dominant.
Finally, by assuming: in Theorem 1 we obtain the following result: Corollary 4: Let q,q(z) ≠ 0 be a univalent function in U and be analytic in U satisfy (2). If the subordination Eq. 7: holds then and q is the best dominant.
Note that Corollary 4, gives sufficient conditions for functions f∈A to be in the class H( ,1,1, (z); (z)).

α Φ Ψ
An application of Theorem 1, next result shows the sufficient conditions for functions f∈A to be in the class H( , , , (z); (z)).
Theorem 3: Let q(z) be convex univalent in the unit disk U. Suppose that g ia an analytic in the unit disk such that and q(z) is the best subordinant.