báo cáo hóa học:On quotients and differences of hypergeometric functions

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báo cáo hóa học:"On quotients and differences of hypergeometric functions"Journal of Inequalities andApplications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. On quotients and differences of hypergeometric functions Journal of Inequalities and Applications 2011, 2011:141 doi:10.1186/1029-242X-2011-141 Slavko Simic (ssimic@turing.mi.sanu.ac.rs) Matti Vuorinen (vuorinen@utu.fi) ISSN 1029-242X Article type Research Submission date 15 July 2011 Acceptance date 21 December 2011 Publication date 21 December 2011 Article URL http://www.journalofinequalitiesandapplications.com/content/2011/1/ This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Simic and Vuorinen ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. On quotients and differences of hypergeometric functions Slavko Simi´∗1 and Matti Vuorinen2 c 1 Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia 2 Department of Mathematics, University of Turku, 20014 Turku, Finland *Corresponding author: ssimic@turing.mi.sanu.ac.rs Email address: MV: vuorinen@utu.fi Abstract For Gaussian hypergeometric functions F (x) = F (a, b; c; x), a, b, c > 0, we consider the quotient QF (x, y ) = (F (x) + F (y ))/F (z ) and the difference DF (x, y ) = F (x) + F (y ) − F (z ) for 0 < x, y < 1 with z = x + y − xy. We give best possible bounds for both expressions under various hypotheses about the parameter triple (a, b; c). 2010 Mathematics Subject Classification: 26D06; 33C05. Keywords: sub-additivity; hypergeometric functions; inequalities. 1. Introduction Among special functions, the hypergeometric function has perhaps the widest rangeof applications. For instance, several well-known classes of special functions such ascomplete elliptic integrals, Legendre functions, Chebyshev and Jacobi polynomials, andsome elementary functions, such as the logarithm, are particular cases of it, cf. [1]. Ina recent article [2] the authors studied various extensions of the Bernoulli inequality forfunctions of logarithmic type. In particular, the zero-balanced hypergeometric functionF (a, b; a + b; x), a, b > 0 occurs in these studies, because it has a logarithmic singularityat x = 1, see (2.8) below. We now continue the discussion of some of the questions forquotients and differences of hypergeometric functions that were left open in [2]. 12 Motivated by the asymptotic behavior of the function F (x) = F (a, b; c; x) when x → 1− ,see (2.8), we define for 0 < x, y < 1, a, b, c > 0 F (x) + F (y )(1.1) QF (x, y ) := , DF (x, y ) := F (x) + F (y ) − F (x + y − xy ). F (x + y − xy ) Our task in this article is to give tight bounds for these quotients and differencesassuming various relationships between the parameters a, b, c. For the general case, we can formulate the following theorem.Theorem 1.2. For a, b, c > 0 and 0 < x, y < 1 let QF be as in (1.1). Then,(1.3) 0 < QF (x, y ) ≤ 2. The bounds in (1.3) are best possible as can be seen by taking [1, 15.1.8] F (x) = F (a, b; b; x) = (1 − x)−a := F0 (x). Then, (1 − x)−a + (1 − y )−a (1 − x)−a + (1 − y )−a = (1 − x)a + (1 − y )a , QF0 (x, y ) = = (1 − x − y + xy )−a ((1 − x)(1 − y ))−aand the conclusion follows immediately. Similarly,Theorem 1.4. For a, b > 0, c > a + b and 0 < x, y < 1, we have(1.5) |DF (x, y )| ≤ A, Γ(c)Γ(c−a−b)with A = A(a, b, c) = = F (a, b; c; 1) as the best possible constant. Γ(c−a)Γ(c−b) Most intriguing is the zero-balanced case. For example, 3Theorem 1.6. For a, b > 0 and 0 < x, y < 1 let DF be as in (1.1). Then, R(1.7) ...