A1 Refereed original research article in a scientific journal
On hypergeometric functions and function spaces
Authors: Balasubramanian R., Ponnusamy S., Vuorinen M.
Publication year: 2002
Journal: Journal of Computational and Applied Mathematics
Journal name in source: Journal of Computational and Applied Mathematics
Volume: 139
Issue: 2
First page : 299
Last page: 322
Number of pages: 24
ISSN: 0377-0427
DOI: https://doi.org/10.1016/S0377-0427(01)00417-4
Web address : http://api.elsevier.com/content/abstract/scopus_id:0037083009
Abstract
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 △ and the subspace R(β) = {f ∈ A: ∃ η ∈ R such that Re e (f′ (z) - β >0, z ∈ △ }. For f ∈ A, we consider integral transforms of the type f (tz) Vλ (f) = ∫ λ (t)t/f(tz) dt, where λ(t) is a real valued nonnegative weight function normalized so that ∫ λ(t) = 1. We obtain conditions on β and the function λ such that V(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. © 2002 Elsevier Science B.V. All rights reserved.
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 △ and the subspace R(β) = {f ∈ A: ∃ η ∈ R such that Re e (f′ (z) - β >0, z ∈ △ }. For f ∈ A, we consider integral transforms of the type f (tz) Vλ (f) = ∫ λ (t)t/f(tz) dt, where λ(t) is a real valued nonnegative weight function normalized so that ∫ λ(t) = 1. We obtain conditions on β and the function λ such that V(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. © 2002 Elsevier Science B.V. All rights reserved.
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