Admissible classes of multivalent functions associated with an integral operator

In this paper we investigate some applications of the differential subordination and superordination of classes of admissible functions associated with an integral operator. Additionally, differential sandwich-type results are obtained.


Introduction.
Let H(U) be the class of functions analytic in the disk U = {z ∈ C : |z| < 1} and H[a, n] be the subclass of H (U) consisting of functions of the form: f (z) = a + a n z n + a n+1 z n+1 + . . . .
Let f and F be members of H(U), the function f (z) is said to be subordinate to F (z), or F (z) is said to be superordinate to f (z), if there exists a function ω(z) analytic in U with ω(0) = 0 and |ω(z)| < 1, z ∈ U, such that f (z) = F (ω(z)). In such a case we write f (z) ≺ F (z). If F is univalent, then f (z) ≺ F (z) if and only if f (0) = F (0) and f (U) ⊂ F (U) (see [7,10] and [11]).
Let φ : C 3 × U → C and h(z) be univalent in U. If p(z) is analytic in U and satisfies the second order differential subordination: (1.1) φ p(z), zp (z), z 2 p (z); z ≺ h(z), then p(z) is a solution of the differential subordination (1.1). The univalent function q(z) is called a dominant of the solutions of (1.1) if p(z) ≺ q(z) for all p (z) satisfying (1.1). A univalent dominantq that satisfiesq ≺ q for all dominants of (1.1) is called the best dominant. If p(z) and φ p(z), zp (z), z 2 p (z); z are univalent in U and if p(z) satisfies second order differential superordination: then p(z) is a solution of the differential superordination (1.2). An analytic function q(z) is called a subordinant of the solutions of (1.2) if q(z) ≺ p(z) for all p(z) satisfying (1.2). A univalent subordinantq that satisfies q ≺q for all subordinants of (1.2) is called the best subordinant. Let A(p) denote the class of all analytic functions and p-valent of the form: Motivated essentially by Jung et al. [9], Shams et al. [12] introduced the integral operator I α p : A(p) → A(p) as follows: For f ∈ A (p) given by (1.3), then from (1.4), we deduce that It is easily verified from (1.5) that . We note that the integral operator I α 1 = I α was defined by Jung et al. [9]. To prove our results, we need the following definitions and lemmas. Denote by F the set of all functions q that are analytic and injective on and are such that q (ζ) = 0 for ζ ∈ ∂U \ E(q). Further let the subclass of F for which q(0) = a be denoted by F(a) and F(0) ≡ F 0 .
In order to prove our results, we shall make use of the following classes of admissible functions. Let Ω be a set in C, q ∈ F and n be a positive integer. The class of admissible functions Ψ n [Ω, q], consists of those functions ψ : C 3 × U → C that satisfy the admissibility condition ψ(r, s, t; z) / ∈ Ω whenever r = q(ζ), s = kζq (ζ), t s where z ∈ U, ζ ∈ ∂U \ E(q) and k ≥ n. We write Ψ 1 [Ω, q] as Ψ[Ω, q].
In our investigation we need the following lemmas which are proved by Miller and Mocanu [10] and [11].
If the analytic function g(z) = a + a n z n + a n+1 z n+1 + . . . satisfies In particular, Aouf and Seoudy [6] investigated a subordination and superordination problems for multivalent functions defined by the integral operator I α p , they have determined classes of admissible functions so that where q 1 and q 2 are given univalent functions in U.
In this paper, we determine the sufficient conditions for certain classes of admissible functions of multivalent functions associated with I α p so that and where µ > 0 and q 1 and q 2 are given univalent functions in U. Additionally, differential sandwich-type results are obtained. A similar problem for analytic functions was studied by Aghalary et al. [1], Ali et al. [2], Aouf et al. [4], and Kim and Srivastava [8] and others (see [3,5] and [6]).

Subordination results involving
Unless otherwise mentioned, we assume throughout this paper that α > 2, µ > 0, p ∈ N, z ∈ U and all powers are principal ones.
Proof. Define the analytic function g(z) in U by In view of the relation (1.6), from (2.2) we get Further computations show that Define the transformations from C 3 to C by Using (2.2)-(2.6), we obtain Hence (2.1) becomes ψ g(z), zg (z), z 2 g (z); z ∈ Ω.
The proof is completed if it can be shown that the admissibility condition for φ ∈ Φ 1 [Ω, q, µ] is equivalent to the admissibility condition for ψ given in Definition 1.1. Note that t s and hence ψ ∈ Ψ µp [Ω, q]. By Lemma 1.3, If Ω = C is a simply connected domain, then Ω = h(U) for some conformal mapping h of U onto Ω. In this case the class Φ 1 [h(U), q, µ] is written as Φ 1 [h, q, µ]. The following result is an immediate consequence of Theorem 2.2.
Our next result is an extension of Theorem 2.2 to the case where the behavior of q on ∂U is not known.

Proof. Theorem 2.2 yields
The result is now deduced from q ρ (z) ≺ q(z).
The next theorem yields the best dominant of the differential subordination (2.8).
Theorem 2.6. Let h be univalent in U. Let φ : C 3 × U → C. Suppose that the differential equation has a solution q such that q(0) = 0 and satisfies one of the following conditions: q is univalent in U and φ ∈ Φ 1 [h, q ρ , µ], for some ρ ∈ (0, 1), or and q is the best dominant.
Proof. Following the same arguments in [10, Theorem 2.3e, p. 31], we deduce that q is a dominant from Theorems 2.3 and 2.5. Since q satisfies (2.9) it is also a solution of (2.8) and therefore q will be dominated by all dominants. Hence q is the best dominant.

Definition 2.7.
Let Ω be a set in C and M > 0. The class of admissible functions Φ 1 [Ω, M, µ] consists of those functions φ : kM for all real θ and k ≥ µp.
The proof is completed if it can be shown that the admissibility condition for φ ∈ Φ 2 [Ω, q, µ] is equivalent to the admissibility condition for ψ given in Definition 1.1. Note that and hence ψ ∈ Ψ µ [Ω, q]. By Lemma 1.3, g(z) ≺ q(z) or I α p f (z) z p−1 µ ≺ q(z).
Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 2.14. ,