Báo cáo toán học: Subclasses of Uniformly Starlike and Convex Functions Defined by Certain Integral Operator

báo cáo này, chúng ta xem xét một lớp học của các chức năng starlike thống nhất được định nghĩa bởi nhà điều hành tách rời nhất định. Chúng tôi xác định một điều kiện đủ cho một hàm f được thống nhất starlike chức năng đó cũng là cần thiết khi e có hệ số tiêu cực.
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Báo cáo toán học: " Subclasses of Uniformly Starlike and Convex Functions Defined by Certain Integral Operator" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:3 (2005) 319–334 RI 0$7+(0$7,&6 ‹ 9$67 Subclasses of Uniformly Starlike and Convex Functions Defined by Certain Integral Operator Maslina Darus1 , Aini Janteng2 , and Suzeini Abdul Halim2 1 School of Mathematical Sciences, Faculty of Sciences and Technology, University Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia 2 Institute of Mathematical Sciences, University Malaya 50603 Kuala Lumpur, Malaysia Received July 21, 2004 Revised March 2, 2005Abstract. In this paper, we consider a class of uniformly starlike functions definedby certain integral operator. We determine a sufficient condition for a function f tobe uniformly starlike function that is also necessary when f has negative coefficients.Similar results for corresponding subclasses of uniformly convex functions are alsoobtained.1. IntroductionLet S denote the class of functions f which are analytic and univalent in D ={z : 0 < |z | < 1} and given by ∞ an z n , an ≥ 0 . f (z ) = z + (1) n=2A function f ∈ S is called a uniformly starlike function if and only if z f (z ) z f (z ) ≥ −1 , z ∈ D. Re f (z ) f (z )We denote this class by Sp . A function f ∈ S is called a uniformly convex function if and only if320 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim zf (z ) z f (z ) ≥ , z ∈ D. Re 1 + f (z ) f (z )We denote this class by UCV . Rønning [3] generalized the class Sp and UCV by introducing a parameter αin the following way.Definition 1. [4] A function f ∈ Sp (α), 0 α 1, if f satisfies the analyticcharacterization z f (z ) z f (z ) −1 −α Re f (z ) f (z )and f ∈ UCV (α) if and only if zf ∈ Sp (α). In [1], Bharati et al. obtained coefficient characterization for some subclassesof Sp (α) and UCV (α).Definition 2. [5] Let T be the subclass of S consisting of functions f of theform ∞ an z n , an ≥ 0 . f (z ) = z − (2) n=2 Also, Bharati et al. in [1] obtained coefficient characterization for somesubclasses of Sp (α) and UCV (α) for f ∈ T . Recently, Jung et al. [2] introduced the following one-parameter families ofintegral operator for functions f ∈ S : z α−1 α+β α t Qα f (z ) tβ −1 f (t)dt, (α > 0, β > −1) 1− = (3) β zβ β z 0and z α+1 tα−1 f (t)dt, (α > −1). Jα f (z ) = (4) zα 0They showed that ∞ Γ(α + β + 1) Γ(β + n) Qα f (z ) = z + an z n , (α > 0, β > −1) ...