Differential sandwich theorems for some subclasses of analytic functions associated with linear operator
Introduction
Let be the class of functions analytic in Δ := {z:∣z∣ < 1} and be the subclass of consisting of functions of the form f(z) = a + anzn + an+1zn+1 + ⋯ . Let be the subclass of consisting of functions of the form f(z) = z + a2z2 + ⋯ . Let and let . If p and ϕ(p(z), zp′(z), z2p″(z);z) are univalent and if p satisfies the second order superordinationthen p is a solution of the differential superordination (1.1). (If f is subordinate to F, then F is called a superordinate of f.) An analytic function q is called a subordinant if q ≺ p for all p satisfying (1.1). An univalent subordinant that satisfies q ≺ q for all subordinants q of (1.1) is said to be the best subordinant. Recently Miller and Mocanu [5] obtained conditions on h, q and ϕ for which the following implication holds:Using the results of Miller and Mocanu [5], Bulboaca [2] considered certain classes of first order differential superordinations as well as superordination-preserving operators [1]. Using the results of [2], Shanmugam et al. [6] obtained sufficient conditions for a normalized analytic function f(z) to satisfyandrespectively where q1 and q2 are given univalent functions in Δ. Also, Tuneski [7] obtained a sufficient condition for starlikeness of the quantity . In the present investigation, we obtain sufficient conditions for the normalized analytic function f(z) to satisfyAlso we obtain results using Carlson–Shaffer operator, Ruscheweyh and Sălăgean derivatives.
Let the function ϕ(a, c; z) be given bywhere (x)n is the Pochhammer symbol defined byCorresponding to the function ϕ(a, c; z), Carlson and Shaffer [3] introduced a linear operator L(a, c), which is defined by the following Hadamard product (or convolution):We note thatwhere Dδf is the Ruscheweyh derivative of f(z). The Sălăgean derivative of a function f(z), denoted by is denoted byAlso, and .
Section snippets
Preliminaries
In our present investigation, we shall need the following definition and results. In this paper unless otherwise mentioned α, β, δ are complex numbers. Definition 1 Denote by Q, the set of all functions f that are analytic and injective on , whereand are such that f′(ζ)≠0 for ζ ∈ ∂ Δ − E(f). Theorem 1 Let q be univalent in the unit disk Δ and θ and ϕ be analytic in a domain D containing q(Δ) with ϕ(ω)≠0 when ω ∈ q(Δ).[5, Definition 2, p. 817]
[4, Theorem 3.4 h, p. 132]
Set ξ(z) = zq′(z)ϕ(q(z)), h(z) = θ(q(z)) + ξ(z). Suppose that,
- 1.
ξ(z) is starlike
Subordination and superordination for analytic functions
By using Lemma 1 we first prove the following. Theorem 3 Let q be univalent in Δ with q(0) = 1, and satisfyingLetIf satisfiesthenand q is the best dominant. Proof Define the function p(z) byThen a computation shows thatNow (3.3) becomesand Theorem 3 follows as an application of Lemma 1. □ Corollary 1 Let
Applications to Carlson–Shaffer operator
Theorem 7 Let q be univalent in Δ with q(0) = 1. Further assuming that (3.1) holds. LetIf , satisfiesthenand q is the best dominant. Proof Define the function p(z) byBy taking logarithmic derivative of (4.1), we getBy using the identity,
Applications to Sãlãgean derivative
Theorem 12 Let q be univalent in Δ with q(0) = 1. Further assuming that (3.1) holds. LetIf , satisfiesthenand q is the best dominant. Proof Define the function p(z) byBy taking logarithmic derivative of p(z) given by (5.3) we getBy using the identity, , and (5.3) in (5.4) we obtain . Then (5.2)
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