# tailieunhanh - 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: 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 . Miller and . Mocanu and the concept of strong diﬀerential subordination was introduced in [1], [2] by . 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 .

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