View More View Less
  • 1 Babeş-Bolyai University Faculty of Economics RO-400591 Cluj-Napoca Romania
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

In this paper we present some Turán type inequalities for the probability density function (pdf) of the non-central chi-squared distribution, non-central chi distribution and Student distribution, respectively. Moreover, we improve a result of Laforgia and Natalini concerning a Turán type inequality for the modified Bessel functions of the second kind.

  • Abreu, L. D. and Bustoz, J. , Turán inequalities for symmetric Askey-Wilson polynomials, Rocky Mountain Journal of Mathematics , 30 (2000), no. 2, 401–409. MR 2002c :33006

    Bustoz J. , 'Turán inequalities for symmetric Askey-Wilson polynomials ' (2009 ) 30 Rocky Mountain Journal of Mathematics : 401 -409.

    • Search Google Scholar
  • Alzer, H., Gerhold, S., Kauers, M. and Lupaş, A. , On Turán’s inequality for Legendre polynomials, Expositiones Mathematicae , 25 (2007), no. 2, 181–186. MR 2008e :33012

    Lupaş A. , 'On Turán’s inequality for Legendre polynomials ' (2007 ) 25 Expositiones Mathematicae : 181 -186.

    • Search Google Scholar
  • Amos, D. E. , Computation of modified Bessel functions and their ratios, Mathematics of Computation , 28 (1974), 239–251. MR 48 #11612

    Amos D. E. , 'Computation of modified Bessel functions and their ratios ' (1974 ) 28 Mathematics of Computation : 239 -251.

    • Search Google Scholar
  • András, Sz. and Baricz, Á. , Properties of the probability density function of the non-central chi-squared distribution, Journal of Mathematical Analysis and Applications , 346 (2008), no. 2, 395–402. MR 2431535

    Baricz , 'Properties of the probability density function of the non-central chi-squared distribution ' (2008 ) 346 Journal of Mathematical Analysis and Applications : 395 -402.

    • Search Google Scholar
  • Baricz, Á. , Turán type inequalities for generalized complete elliptic integrals, Mathematische Zeithschrift , 256 (2007), no. 4, 895–911. MR 2008f :26016

    Baricz , 'Turán type inequalities for generalized complete elliptic integrals ' (2007 ) 256 Mathematische Zeithschrift : 895 -911.

    • Search Google Scholar
  • Baricz, Á. , Functional inequalities for Galué’s generalized modified Bessel functions, Journal of Mathematical Inequalities , 1 (2) (2007), 183–193.

    Baricz , 'Functional inequalities for Galué’s generalized modified Bessel functions ' (2007 ) 1 Journal of Mathematical Inequalities : 183 -193.

    • Search Google Scholar
  • Baricz, Á. , Turán type inequalities for hypergeometric functions, Proceedings of the American Mathematical Society , 136 (2008), no. 9, 3223–3229. MR 2009e :33008

    Baricz , 'Turán type inequalities for hypergeometric functions ' (2008 ) 136 Proceedings of the American Mathematical Society : 3223 -3229.

    • Search Google Scholar
  • Baricz, Á. , Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expositiones Mathematicae , 26 (2008), no. 3, 279–293. MR 2437098

    Baricz , 'Functional inequalities involving Bessel and modified Bessel functions of the first kind ' (2008 ) 26 Expositiones Mathematicae : 279 -293.

    • Search Google Scholar
  • Baricz, Á. and Neuman, E. , Inequalities involving modified Bessel functions of the first kind II, Journal of Mathematical Analysis and Applications , 332 (2007), no. 1, 265–271. MR 2008i :33020

    Neuman E. , 'Inequalities involving modified Bessel functions of the first kind II ' (2007 ) 332 Journal of Mathematical Analysis and Applications : 265 -271.

    • Search Google Scholar
  • Bustoz, J. and Ismail, M. E. H. , Turán inequalities for ultraspherical and continuous q - ultraspherical polynomials, SIAM Journal of Mathematical Analysis , 14 (1983), no. 4, 807–818. MR 85c :33005

    Ismail M. E. H. , 'Turán inequalities for ultraspherical and continuous q- ultraspherical polynomials ' (1983 ) 14 SIAM Journal of Mathematical Analysis : 807 -818.

    • Search Google Scholar
  • Bustoz, J. and Ismail, M. E. H. , Turán inequalities for symmetric orthogonal polynomials, International Journal of Mathematics and Mathematical Sciences , 20 (1997), no. 1, 1–7. MR 98f :33014

    Ismail M. E. H. , 'Turán inequalities for symmetric orthogonal polynomials ' (1997 ) 20 International Journal of Mathematics and Mathematical Sciences : 1 -7.

    • Search Google Scholar
  • Bustoz, J. and Pyung, I. S. , Determinant inequalities for sieved ultraspherical polynomials, International Journal of Mathematics and Mathematical Sciences , 25 (2001), no. 11, 745–751. MR 2002m :33009

    Pyung I. S. , 'Determinant inequalities for sieved ultraspherical polynomials ' (2001 ) 25 International Journal of Mathematics and Mathematical Sciences : 745 -751.

    • Search Google Scholar
  • Bustoz, J. and Savage, N. , Inequalities for ultraspherical and Laguerre polynomials, SIAM Journal on Mathematical Analysis , 10 (1979), no. 5, 902–912. MR 80k :33009

    Savage N. , 'Inequalities for ultraspherical and Laguerre polynomials ' (1979 ) 10 SIAM Journal on Mathematical Analysis : 902 -912.

    • Search Google Scholar
  • Csordas, G., Norfolk, T. S. and Varga, R. S. , The Riemann hypothesis and Turán inequalities, Transactions of the American Mathematical Society , 296 (1986), no. 2, 521–541. MR 87i :11109

    Varga R. S. , 'The Riemann hypothesis and Turán inequalities ' (1986 ) 296 Transactions of the American Mathematical Society : 521 -541.

    • Search Google Scholar
  • Csordas, G. and Williamson, J. , On polynomials satisfying a Turán type inequality, Proceedings of the American Mathematical Society , 43 (1974), no. 2, 367–372. MR 49 #3251

    Williamson J. , 'On polynomials satisfying a Turán type inequality ' (1974 ) 43 Proceedings of the American Mathematical Society : 367 -372.

    • Search Google Scholar
  • Constantinescu, E. , On the inequality of P. Turán for Legendre polynomials, Journal of Inequalities in Pure and Applied Mathematics , 6 (2005), no. 2, Article 28, 4 pp. MR 2006a :33007

  • Danese, A. , Some identities and inequalities involving ultraspherical polynomials, Duke Mathematical Journal , 26 (1959), 349–359. MR 21 #3594

    Danese A. , 'Some identities and inequalities involving ultraspherical polynomials ' (1959 ) 26 Duke Mathematical Journal : 349 -359.

    • Search Google Scholar
  • Devroye, L. , Simulating Bessel random variables, Statistics and Probability Letters , 57 (2002), no. 3, 249–257. MR 2003f :65014

    Devroye L. , 'Simulating Bessel random variables ' (2002 ) 57 Statistics and Probability Letters : 249 -257.

    • Search Google Scholar
  • Dragomir, S. S., Agarwal, R. P. and Barnett, N. S. , Inequalities for beta and gamma functions via some classical and new integral inequalities, Journal of Inequalities and Applications , 5 (2000), no. 2, 103–165. MR 2000m :26012

    Barnett N. S. , 'Inequalities for beta and gamma functions via some classical and new integral inequalities ' (2009 ) 5 Journal of Inequalities and Applications : 103 -165.

    • Search Google Scholar
  • Elbert, Á. and Laforgia, A. , Some monotonicity properties for the zeros of ultraspherical polynomials, Acta Mathematica Hungarica , 48 (1986), no. 1–2, 155–159. MR 88a :33016

    Laforgia A. , 'Some monotonicity properties for the zeros of ultraspherical polynomials ' (1986 ) 48 Acta Mathematica Hungarica : 155 -159.

    • Search Google Scholar
  • Elbert, Á. and Laforgia, A. , Monotonicity results on the zeros of generalized Laguerre polynomials, Journal of Approximation Theory , 51 (1987), no. 2, 168–174. MR 88i :33014

    Laforgia A. , 'Monotonicity results on the zeros of generalized Laguerre polynomials ' (1987 ) 51 Journal of Approximation Theory : 168 -174.

    • Search Google Scholar
  • Gasper, G. , On the extension of Turán’s inequality for Jacobi polynomials, Duke Mathematical Journal , 38 (1971), 415–428. MR 43 #2258

    Gasper G. , 'On the extension of Turán’s inequality for Jacobi polynomials ' (1971 ) 38 Duke Mathematical Journal : 415 -428.

    • Search Google Scholar
  • Gasper, G. , An inequality of Turán type for Jacobi polynomials, Proceedings of the American Mathematical Society , 32 (1972), 435–439. MR 44 #7013

    Gasper G. , 'An inequality of Turán type for Jacobi polynomials ' (1972 ) 32 Proceedings of the American Mathematical Society : 435 -439.

    • Search Google Scholar
  • Giordano, C., Laforgia, A. and Pecărić, J. , Supplement to known inequalities for some special functions, Journal of Mathematical Analysis and Applications , 200 (1996), no. 1, 34–41. MR 97b :33004

    Pecărić J. , 'Supplement to known inequalities for some special functions ' (1996 ) 200 Journal of Mathematical Analysis and Applications : 34 -41.

    • Search Google Scholar
  • Ismail, M. E. H. and Laforgia, A. , Monotonicity properties of determinants of special functions, Constructive Approximation , 26 (2007), no. 1, 1–9. MR 2008e :33001

    Laforgia A. , 'Monotonicity properties of determinants of special functions ' (2007 ) 26 Constructive Approximation : 1 -9.

    • Search Google Scholar
  • Ismail, M. E. H. and Muldoon, M. E. , Monotonicity of the zeros of a cross-product of Bessel functions, SIAM Journal on Mathematical Analysis , 9 (1978), no. 4, 759–767. MR 58 #6388

    Muldoon M. E. , 'Monotonicity of the zeros of a cross-product of Bessel functions ' (1978 ) 9 SIAM Journal on Mathematical Analysis : 759 -767.

    • Search Google Scholar
  • Johnson, N. L., Kotz, S. and Balakrishnan, N. , Continuous Univariate Distributions , vol. 2, 2nd ed., Wiley Interscience, 1995. MR 96j :62029

  • Johnson, O. and Goldschmidt, C. , Preservation of log-concavity on summation, ESAIM Probability and Statistics , 10 (2006), 206–215. (electronic) MR 2007g :60022

    Goldschmidt C. , 'Preservation of log-concavity on summation ' (2006 ) 10 ESAIM Probability and Statistics : 206 -215.

    • Search Google Scholar
  • Joshi, C. M. and Bissu, S. K. , Some inequalities of Bessel and modified Bessel functions, Journal of the Australian Mathematical Society (Series A) , 50 (1991), no. 2, 333–342. MR 92d :33007

    Bissu S. K. , 'Some inequalities of Bessel and modified Bessel functions ' (1991 ) 50 Journal of the Australian Mathematical Society (Series A) : 333 -342.

    • Search Google Scholar
  • Laforgia, A. , Sturm theory for certain class of -Sturm-Liouville equations and Turánians and Wronskians for the zeros of derivative of Bessel functions, Nederl. Akad. Wetensch. Indag. Math. , 44 (1982), no. 3, 295–301. MR 84c :33009

    Laforgia A. , 'Sturm theory for certain class of -Sturm-Liouville equations and Turánians and Wronskians for the zeros of derivative of Bessel functions ' (1982 ) 44 Nederl. Akad. Wetensch. Indag. Math. : 295 -301.

    • Search Google Scholar
  • Laforgia, A. and Natalini, P. , Turán type inequalities for some special functions, Journal of Inequalities in Pure and Applied Mathematics , 7 (2006), no. 1, Article 32, 3 pp. MR 2006m :26028

  • Laforgia, A. and Natalini, P. , On some Turán-type inequalities, Journal of Inequalities and Applications , 2006 , Article 29828, 6 p. (2006). MR 2007c :26028

  • Lorch, L. , Turánians and Wronskians for the zeros of Bessel functions, SIAM Journal on Mathematical Analysis , 11 (1980), no. 2, 223–227. MR 80m :33017

    Lorch L. , 'Turánians and Wronskians for the zeros of Bessel functions ' (1980 ) 11 SIAM Journal on Mathematical Analysis : 223 -227.

    • Search Google Scholar
  • Mitrinović, D. S. , Analytic inequalities , Springer-Verlag, Berlin, 1970. MR 43 #448

    Mitrinović D. S. , '', in Analytic inequalities , (1970 ) -.

  • Muldoon, M. E. , Convexity properties of special functions and their zeros , in: G. V. Milovanovic (ed.), Recent progress in inequalities, Kluwer Academic Publishers. Math. Appl., Dordrecht, 430 (1998), 309–323. MR 1609967

  • Niculescu, C. P. , A new look at Newton’s inequality, Journal of Inequalities in Pure and Applied Mathematics , 1 (2000), no. 2, Article 17. MR 2001h :26020

  • Qiu, S.-L. and Vuorinen, M. , Some properties of the gamma and psi functions, with application, Mathematics of Computation , 74 (2005), no. 250 (electr.), 723–742. MR 2005i :33002

    Vuorinen M. , 'Some properties of the gamma and psi functions, with application ' (2005 ) 74 Mathematics of Computation : 723 -742.

    • Search Google Scholar
  • Simic, S. , Turán’s inequality for Appell polynomials, Journal of Inequalities and Applications , 2006 , Article 91420, 7 pp. (2006). MR 2007d :26009

  • Szász, O. , Inequalities concerning ultraspherical polynomials and Bessel functions, Proceedings of the American Mathematical Society , 1 (1950), 256–267. MR 11 ,662b

    Szász O. , 'Inequalities concerning ultraspherical polynomials and Bessel functions ' (1950 ) 1 Proceedings of the American Mathematical Society : 256 -267.

    • Search Google Scholar
  • Szász, O. , Identities and inequalities concerning orthogonal polynomials and Bessel functions, Journal d’Analyse Mathematique , 1 (1951), 116–134. MR 13 ,232f

    Szász O. , 'Identities and inequalities concerning orthogonal polynomials and Bessel functions ' (1951 ) 1 Journal d’Analyse Mathematique : 116 -134.

    • Search Google Scholar
  • Szegő, G. , On an inequality of P. Turán concerning Legendre polynomials, Bulletin of the American Mathematical Society , 54 (1948), 401–405. MR 9 ,429d

    Szegő G. , 'On an inequality of P. Turán concerning Legendre polynomials ' (1948 ) 54 Bulletin of the American Mathematical Society : 401 -405.

    • Search Google Scholar
  • Thiruvenkatachar, V. K. and Nanjundiah, T. S. , Inequalities concerning Bessel functions and orthogonal polynomials, Proceedings of the Indian National Science Academy (Part A) , 33 (1951), 373–384. MR 14 ,44b

    Nanjundiah T. S. , 'Inequalities concerning Bessel functions and orthogonal polynomials ' (1951 ) 33 Proceedings of the Indian National Science Academy (Part A) : 373 -384.

    • Search Google Scholar
  • Turán, P. , On the zeros of the polynomials of Legendre, Časopis Pro Pęstování Matematiky, 75 (1950), 113–122. MR 12 ,824g

    Turán P. , 'On the zeros of the polynomials of Legendre ' (1950 ) 75 Časopis Pro Pęstování Matematiky : 113 -122.

    • Search Google Scholar
  • Venkatachaliengar, K. and Lakshmana Rao, S. K. , On Turán’s inequality for ultraspherical polynomials, Proceedings of the American Mathematical Society , 8 (1957), 1075–1087. MR 19 ,954c

    Lakshmana Rao S. K. , 'On Turán’s inequality for ultraspherical polynomials ' (1957 ) 8 Proceedings of the American Mathematical Society : 1075 -1087.

    • Search Google Scholar
  • Watson, G. N. , A Treatise on the Theory of Bessel Functions , Cambridge University Press, Cambridge, 1944. MR 6 ,64a

    Watson G. N. , '', in A Treatise on the Theory of Bessel Functions , (1944 ) -.