On hypergeometric functions and function spaces
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The aim of this paper is to discuss the role of hypergeometric functions in function spaces and to prove some new results for these functions. The first part of this paper proves results such as monotone, convexity and concavity properties of sums of products of hypergeometric functions. The second part of our results deals with the space A of all normalized analytic functions f, f(0)=0=f′(0)−1, in the unit disk Δ and the subspaceR(β)={f∈A:∃η∈RsuchthatReeiη(f′(z)−β)>0,z∈Δ}.For f∈A, we consider integral transforms of the typeVλ(f)=∫01λ(t)f(tz)tdt,where λ(t) is a real valued nonnegative weight function normalized so that ∫01λ(t)=1. We obtain conditions on β and the function λ such that Vλ(f) takes each member of R(β) into a starlike function of order β,β∈[0,1/2]. These results extend and improve the earlier known results in these directions. We end the paper with an open problem.
论文关键词:30C45,33−02,33C05,33C15,Univalent,Starlike,Convex,Close-to-convex,Hypergeometric functions
论文评审过程:Received 2 May 2000, Revised 30 January 2001, Available online 27 November 2001.
论文官网地址:https://doi.org/10.1016/S0377-0427(01)00417-4