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Strong differential subordination

In this paper we study the strong differential subordinations in the general case, following the general theory of differential subordinations presented. | Turk J Math 33 (2009) , 249 – 257. ¨ ITAK ˙ c TUB doi:10.3906/mat-0804-16 Strong differential subordination Georgia Irina Oros and Gheorghe Oros Abstract The concept of differential subordination was introduced in [4] by S. S. Miller and P. T. Mocanu and the concept of strong differential subordination was introduced in [1] by J. A. Antonino and S. Romaguera. This last concept was applied in the special case of Briot-Bouquet strong differential subordination. In this paper we study the strong differential subordinations in the general case, following the general theory of differential subordinations presented in [4]. Key Words: Analytic function, differential subordination, subordination, strong subordination, univalent function. 1. Introduction Let H = H(U ) denote the class of functions analytic in U . For n a positive integer and a ∈ C, let H[a, n] = {f ∈ H; f(z) = a + an z n + an+1 z n+1 + . . . , z ∈ U }. Let A be the class of functions f of the form f(z) = z + a2 z 2 + a3 z 3 + . . . , z ∈ U, which are analytic in the unit disk. Definition 1 [1], [2], [3] Let H(z, ξ) be analytic in U × U and let f(z) analytic and univalent in U . The function H(z, ξ) is strongly subordinate to f(z), written H(z, ξ) ≺≺ f(z) if for ξ ∈ U , the function of z , H(z, ξ) is subordinate to f(z). Remark 1 Since f(z) is analytic and univalent, Definition 1 is equivalent to H(0, ξ) = f(0) and H(U × U ) ⊂ f(U ). (1) 2000 AMS Mathematics Subject Classification: 30C45, 34A30. 249 OROS, OROS 2. Main results Let Ω and Δ be any sets in C, let p be analytic in the unit disk U and let ψ(r, s, t; z, ξ) : C3 ×U ×U → C. As in [4], in this article we consider conditions on Ω, Δ and ψ for which the following implication holds: {ψ(p(z), zp (z), z 2 p (z); z, ξ | z ∈ U, ξ ∈ U } ⊂ Ω ⇒ p(U ) ⊂ Δ. (2) There are three distinct cases to consider in analyzing this implication, which we list as the following problems. Problem 1. Given Ω and Δ , find conditions on the function ψ so that (2) .