On hypergeometric functions and function spaces

Balasubramanian, R. ; Ponnusamy, S. ; Vuorinen, M. (2002) On hypergeometric functions and function spaces Journal of Computational and Applied Mathematics, 139 (2). pp. 299-322. ISSN 0377-0427

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S03770...

Related URL: http://dx.doi.org/10.1016/S0377-0427(01)00417-4


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 subspace R(β) = {f ∈ A:∃ η ∈ R such that Re e(f'(z) - β) > 0. z ∈ Δ}. For f ∈ A, we consider integral transforms of the type Vλ(f) = ∫01λ(t) f(tz)/t dt 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.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Univalent; Starlike; Convex; Close-to-convex; Hypergeometric Functions
ID Code:1296
Deposited On:04 Oct 2010 07:54
Last Modified:16 May 2011 04:35

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