Strong Differential Sandwich Results of λ-Pseudo-Starlike Functions with Respect to Symmetrical Points

In the present investigation, by considering suitable classes of admissible functions, we establish strong differential subordination and superordination properties for λ-pseudo-starlike functions with respect to symmetrical points in the open unit disk U . These results are applied to obtain strong differential sandwich results.


Introduction and Preliminaries
Let H(U ) be the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1}. For a positive integer n and a ∈ C, let 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 + · · · , with H = H [1,1].
Let A stand for the class of all analytic functions in U and having the form: f (z) = z + ∞ n=2 a n z n , (z ∈ U ), A function f ∈ A is called starlike with respect to symmetrical points, if (see [10]) The set of all such functions is denote by S * s . Recently, Babalola [2] defined the class L λ of λ-pseudo-starlike functions as follows: G(z, ζ) ≺≺ f (z) ⇐⇒ G(0, ζ) = f (0) and G(U ×Ū ) ⊂ f (U ). Definition 1.1 ([6]). Let φ : C 3 × U ×Ū −→ C and let h be a univalent function in U . If p is analytic in U and satisfies the following (second-order) strong differential subordination: (1) φ p(z), zp (z), z 2 p (z); z, ζ ≺≺ h(z), then p is called a solution of the strong differential subordination (1). The univalent function q is called a dominant of the solutions of the strong differential subordination or more simply a dominant if p(z) ≺ q(z) for all p satisfying (1). A dominantq that satisfiesq(z) ≺ q(z) for all dominants q of (1) is said to be the best dominant. 7]). Let φ : C 3 ×U ×Ū −→ C and let h be analytic function in U . If p and φ p(z), zp (z), z 2 p (z); z, ζ are univalent in U for ζ ∈Ū and satisfy the following (second-order) strong differential superordination: (2) h(z) ≺≺ φ p(z), zp (z), z 2 p (z); z, ζ , then p is called a solution of the strong differential superordination (2). An analytic function q is called a subordinant of the solutions of the strong differential superordination or more simply a subordinant if q(z) ≺ p(z) for all p satisfying (2). A univalent subordinantq that satisfies q(z) ≺q(z) for all subordinants q of (2) is said to be the best subordinant.

Definition 1.3 ([6]
). Denote by Q the set consisting of all functions q that are analytic and injective onŪ \E(q), where and are such that q (ξ) = 0 for ξ ∈ ∂U \E(q). Furthermore, let the subclass of Q for which q(0) = a be denoted by Q(a), Q(0) ≡ Q 0 and Q(1) ≡ Q 1 .
In our investigations, we will need the following lemmas: In recent years, several authors obtained many interesting results in strong differential subordination and superordination [1,3,4,11,12,13]. In this work, by making use of the strong differential subordination results and strong differential superordination results of Oros and Oros [8,9], we introduce and study certain suitable classes of admissible functions and derive some strong differential subordination and superordination properties of λpseudo-starlike functions with respect to symmetrical points.
Proof. We define the function p by .
It is clear that p is analytic in U . Simple calculations from (4), we obtain Further computations show that Define the transforms from C 3 to C by The proof shall make use of Lemma 1.1. Using equations (4), (5) and (6), it follows from (7) that To complete the proof, we next show that the admissibility condition for φ ∈ Φ L [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.4. Note that t s Hence ψ ∈ Ψ [Ω, q]. By Lemma 1.1, p(z) ≺ q(z) or equivalently We consider the special situation when Ω = C is a simply connected domain. In this case Ω = h(U ), for some conformal mapping h of U onto Ω and the class Φ L [h(U ), q] is written as Φ L [h, q]. The following result is an immediate consequence of Theorem 2.1.
By taking φ(u, v, w; z, ζ) = u + v βu+γ , (β, γ ∈ C) in Theorem 2.2, we state the following corollary: Corollary 2.1. Let β, γ ∈ C and let h be convex in U with h(0) = 1 and The next result is an extension of Theorem 2.1 to the case where the behavior of q on ∂U is not known.
By using Theorem 2.1 and the comment associated with where w is any function mapping U into U , with w(z) = ρz, we obtain p ρ (z) ≺ q ρ (z) for ρ ∈ (ρ 0 , 1). By letting ρ → 1 − , we get p(z) ≺ q(z). Therefore The next result gives the best dominant of the strong differential subordination (9): Theorem 2.4. Let h be univalent in U and φ : C 3 × U ×Ū −→ C. Suppose that the differential equation has a solution q with q(0) = 1 and satisfies one of the following conditions: q is univalent in U and φ ∈ Φ L [h, q ρ ] for some ρ ∈ (0, 1), (3) q is univalent in U and there exists ρ 0 ∈ (0, 1) such that φ ∈ Φ L [h ρ , q ρ ] for all ρ ∈ (ρ 0 , 1). If f ∈ A satisfies (9), then and q is the best dominant.
Proof. By applying Theorem 2.2 and Theorem 2.3, we deduce that q is a dominant of (9). Since q satisfies (10), it is also a solution of (9) and therefore q will be dominated by all dominants. Hence q is the best dominant of (9). In M for all θ and k ≥ 1. , When Ω = q(U ) = {w : |w − 1| < M }, the class Φ L [Ω, M ] is simply denoted by Φ L [M ], then corollary 2.3 takes the following form: , This implication follows from Corollary 2.4 by taking φ(u, v, w; z, ζ) = w − v + 2.