Proceedings of the American Mathematical Society

Author(s): Árpád Baricz
Journal: Proc. Amer. Math. Soc. 136 (2008), 3223-3229.
MSC (2000): Primary 33C05; Secondary 26D07
Posted: April 29, 2008
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Abstract: In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems.

1.: H. Alzer, and C. Berg, Some classes of completely monotonic functions, II, Ramanujan J. 11 (2006), 225-248. MR 2267677 (2007k:33001)
2.: H. Alzer, S. Gerhold, M. Kauers, and A. Lupaş, On Turán's inequality for Legendre polynomials, Expo. Math. 25 (2007), 181-186. MR 2319577
3.: G.D. Anderson, S.L. Qiu, M.K. Vamanamurthy and M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math. 192 (2000), 1-37. MR 1741031 (2001b:33027)
4.: G.E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
5.: Á. Baricz, Turán type inequalities for generalized complete elliptic integrals, Math. Z. 256 (2007), 895-911. MR 2308896
6.: Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. (2008), doi:10.1016/j.exmath.2008.01.001.
7.: C. Berg and G. Forst, Potential Theory and Locally Compact Abellian Groups, Springer, Berlin, 1975. MR 0481057 (58:1204)
8.: J. Bustoz and M.E.H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), 659-667. MR 856710 (87m:33002)
9.: G. Csordas and J. Williamson, On polynomials satisfying a Turán type inequality, Proc. Amer. Math. Soc. 43(2) (1974), 367-372. MR 0338487 (49:3251)
10.: G. Gasper, On the extension of Turán's inequality to Jacobi polynomials, Duke Math. J. 38 (1971), 415-428. MR 0276514 (43:2258)
11.: G. Gasper, An inequality of Turán type for Jacobi polynomials, Proc. Amer. Math. Soc. 32 (1972), 435-439. MR 0289826 (44:7013)
12.: V. Heikkala, M.K. Vamanamurthy and M. Vuorinen, Generalized elliptic integrals, Comput. Methods Funct. Theory 9 (1) (2009), 75-109.
13.: C.M. Joshi and S.K. Bissu, Some inequalities of Bessel and modified Bessel functions, J. Austral. Math. Soc. (Series A) 50 (1991), 333-342. MR 1094928 (92d:33007)
14.: S. Karlin, G. Szegő, On certain determinants whose elements are orthogonal polynomials, J. d'Analyse Math. 8 (1960/61), 1-157, reprinted in R. Askey, ed., ``Gábor Szegő Collected Papers'', vol. 3, Birkhäuser, Boston, 1982, 605-761. MR 0142972 (26:539)
15.: O. Szász, Inequalities concerning ultraspherical polynomials and Bessel functions, Proc. Amer. Math. Soc. 1 (1950), 256-267. MR 0034902 (11:662b)
16.: G. Szegő, On an inequality of P. Turán concerning Legendre polynomials, Bull. Amer. Math. Soc. 54 (1948), 401-405. MR 0023954 (9:429d)
17.: G. Szegő, An inequality for Jacobi polynomials, Studies in Math. Anal. and Related Topics, Stanford Univ. Press., Stanford, Calif. (1962), 392-398. MR 0145122 (26:2657)
18.: V.K. Thiruvenkatachar and T.S. Nanjundiah, Inequalities concerning Bessel functions and orthogonal polynomials, Proc. Indian Nat. Acad. Part A 33 (1951), 373-384. MR 0048635 (14:44b)
19.: P. Turán, On the zeros of the polynomials of Legendre, Časopis Pest. Mat. Fys. 75 (1950), 113-122. MR 0041284 (12:824g)
20.: D.V. Widder, The Laplace Trasform, Princeton Univ. Press, NJ, 1941. MR 0005923 (3:232d)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33C05, 26D07

Árpád Baricz
Affiliation: Faculty of Economics, Babes-Bolyai University, RO-400591 Cluj-Napoca, Romania
Email: bariczocsi@yahoo.com

DOI: 10.1090/S0002-9939-08-09353-2
PII: S 0002-9939(08)09353-2
Keywords: Hypergeometric function, Legendre polynomial, Jacobi polynomial, Tur\'an inequality.
Received by editor(s): July 23, 2007
Posted: April 29, 2008
Dedicated: Dedicated to the memory of Professor Alexandru Lupas
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2008, American Mathematical Society

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