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    Baricz, Á. Ponnusamy, S. and Singh, S. 2016. Modified Dini functions: monotonicity patterns and functional inequalities. Acta Mathematica Hungarica, Vol. 149, Issue. 1, p. 120.


    Baricz, Árpád 2015. Bounds for Turánians of modified Bessel functions. Expositiones Mathematicae, Vol. 33, Issue. 2, p. 223.


    Lin, Junshan and Santosa, Fadil 2015. Scattering Resonances for a Two-Dimensional Potential Well with a Thick Barrier. SIAM Journal on Mathematical Analysis, Vol. 47, Issue. 2, p. 1458.


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    Baricz, Árpád and Pogány, Tibor K. 2014. Turán determinants of Bessel functions. Forum Mathematicum, Vol. 26, Issue. 1,


    Gaunt, Robert E. 2014. Inequalities for modified Bessel functions and their integrals. Journal of Mathematical Analysis and Applications, Vol. 420, Issue. 1, p. 373.


    Baricz, Árpád and Ismail, Mourad E. H. 2013. Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions. Constructive Approximation, Vol. 37, Issue. 2, p. 195.


    Kalmykov, S.I. and Karp, D.B. 2013. Log-concavity for series in reciprocal gamma functions and applications. Integral Transforms and Special Functions, Vol. 24, Issue. 11, p. 859.


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    Kokologiannaki, Chrysi G. 2012. Bounds for functions involving ratios of modified Bessel functions. Journal of Mathematical Analysis and Applications, Vol. 385, Issue. 2, p. 737.


    Koumandos, Stamatis and Pedersen, Henrik Laurberg 2012. Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers. Mathematische Nachrichten, Vol. 285, Issue. 17-18, p. 2129.


    Baricz, Árpád Ponnusamy, Saminathan and Vuorinen, Matti 2011. Functional inequalities for modified Bessel functions. Expositiones Mathematicae, Vol. 29, Issue. 4, p. 399.


    ×
  • Bulletin of the Australian Mathematical Society, Volume 82, Issue 2
  • October 2010, pp. 254-264

TURÁN TYPE INEQUALITIES FOR MODIFIED BESSEL FUNCTIONS

  • ÁRPÁD BARICZ (a1)
  • DOI: http://dx.doi.org/10.1017/S000497271000002X
  • Published online: 01 April 2010
Abstract
Abstract

In this paper our aim is to deduce some sharp Turán type inequalities for modified Bessel functions of the first and second kinds. Our proofs are based on explicit formulas for the Turánians of the modified Bessel functions of the first and second kinds and on a formula which is related to the infinite divisibility of the Student t-distribution.

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[2]H. Alzer and G. Felder , ‘A Turán-type inequality for the gamma function’, J. Math. Anal. Appl. 350 (2009), 276282.

[3]H. Alzer , S. Gerhold , M. Kauers and A. Lupaş , ‘On Turán’s inequality for Legendre polynomials’, Expo. Math. 25 (2007), 181186.

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[5]Á. Baricz , ‘Turán type inequalities for generalized complete elliptic integrals’, Math. Z. 256(4) (2007), 895911.

[6]Á. Baricz , ‘Functional inequalities involving Bessel and modified Bessel functions of the first kind’, Expo. Math. 26(3) (2008), 279293.

[7]Á. Baricz , ‘Turán type inequalities for hypergeometric functions’, Proc. Amer. Math. Soc. 136(9) (2008), 32233229.

[8]Á. Baricz , ‘Mills’ ratio: monotonicity patterns and functional inequalities’, J. Math. Anal. Appl. 340(2) (2008), 13621370.

[9]Á. Baricz , ‘On a product of modified Bessel functions’, Proc. Amer. Math. Soc. 137(1) (2009), 189193.

[11]R. W. Barnard , M. B. Gordy and K. C. Richards , ‘A note on Turán type and mean inequalities for the Kummer function’, J. Math. Anal. Appl. 349(1) (2009), 259263.

[12]T. H. Gronwall , ‘An inequality for the Bessel functions of the first kind with imaginary argument’, Ann. of Math. (2) 33(2) (1932), 275278.

[13]E. Grosswald , ‘The Student t-distribution of any degree of freedom is infinitely divisible’, Z. Wahrscheinlichkeitstheorie verw. Gebiete 36(2) (1976), 103109.

[14]M. E. H. Ismail , ‘Bessel functions and the infinite divisibility of the Student t-distribution’, Ann. Probab. 5(4) (1977), 582585.

[15]M. E. H. Ismail and A. Laforgia , ‘Monotonicity properties of determinants of special functions’, Constr. Approx. 26 (2007), 19.

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[22]M. E. Muldoon , ‘Convexity properties of special functions and their zeros’, in: Recent Progress in Inequalities, Math. Appl., 430 (ed. G. V. Milovanovic ) (Kluwer Academic Publishers, Dordrecht, 1998), pp. 309323.

[23]R. Penfold , J.-M. Vanden-Broeck and S. Grandison , ‘Monotonicity of some modified Bessel function products’, Integral Transforms Spec. Funct. 18(2) (2007), 139144.

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[27]O. Szász , ‘Inequalities concerning ultraspherical polynomials and Bessel functions’, Proc. Amer. Math. Soc. 1 (1950), 256267.

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[29]G. Szegő , ‘On an inequality of P. Turán concerning Legendre polynomials’, Bull. Amer. Math. Soc. 54 (1948), 401405.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
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