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Sufficient conditions for univalence obtained by using second order linear strong differential subordinations

In this paper we study the second order linear strong differential subordinations. Our results may be applied to deduce sufficient conditions for univalence in the unit disc, such as starlikeness, convexity, alpha-convexity, close-to-convexity respectively. | Turk J Math 34 (2010) , 13 – 20. ¨ ITAK ˙ c TUB doi:10.3906/mat-0810-6 Sufficient conditions for univalence obtained by using second order linear strong differential subordinations Georgia Irina Oros Abstract The concept of differential subordination was introduced in [3] by S.S. Miller and P.T. Mocanu and the concept of strong differential subordination was introduced in [1], [2] by J.A. Antonino and S. Romaguera. In [5] we have studied the strong differential subordinations in the general case and in [6] we have studied the first order linear strong differential subordinations. In this paper we study the second order linear strong differential subordinations. Our results may be applied to deduce sufficient conditions for univalence in the unit disc, such as starlikeness, convexity, alpha-convexity, close-to-convexity respectively. Key Words: Analytic function, differential subordination, strong differential subordination, linear strong differential subordinations, second order linear strong differential subordinations. 1. Introduction Let H = H(U ) denote the class of analytic functions in U . For a positive integer n 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. In addition, we need the classes of convex, alpha-convex, close-to-convex and starlike (univalent) functions given respectively by zf (z) K = f ∈ A; Re + 1 > 0, z ∈ U , f (z) f(z)f (z) = 0, Mα = f ∈ A, z zf (z) zf (z) Re (1 − α) +α 1+ > 0, z ∈ U f(z) f (z) 2000 AMS Mathematics Subject Classification: 30C45, 34A30. 13 OROS C = {f ∈ A, Re f (z) > 0, z ∈ U }, and S ∗ = {f ∈ A, Re zf (z)/f(z) > 0}. In order to prove our main results we use the following definitions and lemmas. Definition 1 [1], [2] 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 .