Abstract
Transkript
Abstract
Hacettepe Journal of Mathematics and Statistics Volume 40 (6) (2011), 829 – 837 SOME CONVEXITY PROPERTIES FOR TWO NEW P -VALENT INTEGRAL OPERATORS Erhan Deniz∗†, Murat Çağlar∗ and Halit Orhan∗ Received 11 : 11 : 2010 : Accepted 09 : 05 : 2011 Abstract In this paper, we define two new general p-valent integral operators in the unit disc U, and obtain the convexity properties of these integral operators of p-valent functions on some classes of β-uniformly p-valent starlike and β-uniformly p-valent convex functions of complexorder. µ As Rz special cases, the convexity properties of the operators 0 f (t) dt t Rz ′ µ and 0 (g (t)) dt are given. Keywords: Analytic functions, Integral operators, β-uniformly p-valent starlike and β-uniformly p-valent convex functions, Complex order. 2000 AMS Classification: Primary 30 C 80. Secondary 30 C 45. 1. Introduction and preliminaries Let Ap denote the class of functions of the form (1.1) f (z) = z p + ∞ X ak z k , (p ∈ N = {1, 2, . . . , }) , k=p+1 which are analytic in the open disc U = {z ∈ C : |z| < 1}. A function f ∈ S∗p (γ, α) is p−valently starlike of complex order γ (γ ∈ C − {0}) and type α (0 ≤ α < p), that is, f ∈ S∗p (γ, α), if it satisfies the following inequality; 1 zf ′ (z) (1.2) ℜ p+ −p > α, (z ∈ U) . γ f (z) ∗Department of Mathematics, Faculty of Science, Atatürk University, TR-25240 Erzurum, Turkey. E-mail: (E. Deniz) edeniz@atauni.edu.tr (M. Çağlar) mcaglar@atauni.edu.tr (H. Orhan) horhan@atauni.edu.tr † Corresponding Author.