×

The monotonicity and convexity for the ratios of modified Bessel functions of the second kind and applications. (English) Zbl 1426.33015

Summary: Let \(K_v(x)\) be the modified Bessel functions of the second kind of order \(v\). We prove that the function \(x\mapsto K_u(x) K_v(x)/K_{(u+v)/2}(x)^2\) is strictly decreasing on \((0,\infty)\). Our study not only involves the Turán type inequalities, log-convexity or log-concavity of \( K_v(x)\), and the conjecture posed by Baricz, but also yields various new results concerning the monotonicity and convexity of the ratios of the modified Bessel functions of the second kind. As applications of our main theorems, some new sharp inequalities involving \(K_v(x)\) are presented, which contain sharp estimates for \(K_v(x)\) and sharp bounds for the ratios \(K_v'(x)/K_v(x)\) and \( K_{v+1}(x) /K_v(x)\).

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
26A48 Monotonic functions, generalizations
39B62 Functional inequalities, including subadditivity, convexity, etc.
26A51 Convexity of real functions in one variable, generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Watson, G. N., A Treatise on the Theory of Bessel Functions, vi+804 pp. (1944), Cambridge University Press, Cambridge, England; The Macmillan Company, New York · Zbl 0849.33001
[2] Laforgia, A.; Natalini, P., Tur\'an-type inequalities for some special functions, JIPAM. J. Inequal. Pure Appl. Math., 7, 1, Article 22, 3 pp. (electronic) pp. (2006) · Zbl 1126.26017
[3] Laforgia, A.; Natalini, P., On some Tur\'an-type inequalities, J. Inequal. Appl. 2006, Art. ID 29828, 6 pp. · Zbl 1095.33002 · doi:10.1155/JIA/2006/29828
[4] Ismail, Mourad E. H.; Laforgia, Andrea, Monotonicity properties of determinants of special functions, Constr. Approx., 26, 1, 1-9 (2007) · Zbl 1117.33004 · doi:10.1007/s00365-005-0627-4
[5] Baricz, {\'A}rp{\'a}d, Tur\'an type inequalities for hypergeometric functions, Proc. Amer. Math. Soc., 136, 9, 3223-3229 (2008) · Zbl 1154.33001 · doi:10.1090/S0002-9939-08-09353-2
[6] Alzer, Horst; Felder, Giovanni, A Tur\'an-type inequality for the gamma function, J. Math. Anal. Appl., 350, 1, 276-282 (2009) · Zbl 1198.33001 · doi:10.1016/j.jmaa.2008.09.053
[7] Baricz, {\'A}rp{\'a}d, On a product of modified Bessel functions, Proc. Amer. Math. Soc., 137, 1, 189-193 (2009) · Zbl 1195.33008 · doi:10.1090/S0002-9939-08-09571-3
[8] Barnard, Roger W.; Gordy, Michael B.; Richards, Kendall C., A note on Tur\'an type and mean inequalities for the Kummer function, J. Math. Anal. Appl., 349, 1, 259-263 (2009) · Zbl 1162.33001 · doi:10.1016/j.jmaa.2008.08.024
[9] Laforgia, Andrea; Natalini, Pierpaolo, Some inequalities for modified Bessel functions, J. Inequal. Appl. 2010, Art. ID 253035, 10 pp. · Zbl 1187.33002 · doi:10.1155/2010/253035
[10] Baricz, {\'A}rp{\'a}d, Tur\'an type inequalities for some probability density functions, Studia Sci. Math. Hungar., 47, 2, 175-189 (2010) · Zbl 1234.62010 · doi:10.1556/SScMath.2009.1123
[11] Kokologiannaki, Chrysi G., Bounds for functions involving ratios of modified Bessel functions, J. Math. Anal. Appl., 385, 2, 737-742 (2012) · Zbl 1275.33028 · doi:10.1016/j.jmaa.2011.07.004
[12] Baricz, {\'A}rp{\'a}d; Pog{\'a}ny, Tibor K., Tur\'an determinants of Bessel functions, Forum Math., 26, 1, 295-322 (2014) · Zbl 1304.33003 · doi:10.1515/form.2011.160
[13] Ismail, Mourad E. H., Bessel functions and the infinite divisibility of the Student \(t\)-distribution, Ann. Probability, 5, 4, 582-585 (1977) · Zbl 0369.60023
[14] Robert, Christian, Modified Bessel functions and their applications in probability and statistics, Statist. Probab. Lett., 9, 2, 155-161 (1990) · Zbl 0686.62021 · doi:10.1016/0167-7152(92)90011-S
[15] Devroye, Luc, Simulating Bessel random variables, Statist. Probab. Lett., 57, 3, 249-257 (2002) · Zbl 1005.65008 · doi:10.1016/S0167-7152(02)00055-X
[16] LushnikovJAS342003 A. A. Lushnikov, J. S. Bhatt, and I. J. Ford, Stochastic approach to chemical kinetics in ultrafine aerosols, J. Aerosol Sci. 34 (2003), 1117-1133.
[17] van Haeringen, H., Bound states for \(r{}^-{}^2\)-like potentials in one and three dimensions, J. Math. Phys., 19, 10, 2171-2179 (1978) · doi:10.1063/1.523574
[18] TanOP32172007 S. Tan and L. Jiao, Multishrinkage: Analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model, Optim. Lett. 32 (2007), no. 17, 2583-2585.
[19] KhazronIEEESPL152008 P. A. Khazron and I. W. Selesnick, Bayesian estimation of Bessel K-form random vectors in AWGN, IEEE Signal Process. Lett. 15 (2008), 261-264.
[20] Ismail, Mourad E. H.; Muldoon, Martin E., Monotonicity of the zeros of a cross-product of Bessel functions, SIAM J. Math. Anal., 9, 4, 759-767 (1978) · Zbl 0388.33005
[21] Baricz, {\'A}rp{\'a}d, Tur\'an type inequalities for modified Bessel functions, Bull. Aust. Math. Soc., 82, 2, 254-264 (2010) · Zbl 1206.33007 · doi:10.1017/S000497271000002X
[22] Segura, Javier, Bounds for ratios of modified Bessel functions and associated Tur\'an-type inequalities, J. Math. Anal. Appl., 374, 2, 516-528 (2011) · Zbl 1207.33009 · doi:10.1016/j.jmaa.2010.09.030
[23] Baricz, {\'A}rp{\'a}d, Bounds for Tur\'anians of modified Bessel functions, Expo. Math., 33, 2, 223-251 (2015) · Zbl 1316.33002 · doi:10.1016/j.exmath.2014.07.001
[24] Baricz, {\'A}rp{\'a}d; Ponnusamy, Saminathan, On Tur\'an type inequalities for modified Bessel functions, Proc. Amer. Math. Soc., 141, 2, 523-532 (2013) · Zbl 1272.33005 · doi:10.1090/S0002-9939-2012-11325-5
[25] Abramowitz, Milton; Stegun, Irene A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, xiv+1046 pp. (1964), For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. · Zbl 0171.38503
[26] Grosswald, E., The Student \(t\)-distribution of any degree of freedom is infinitely divisible, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36, 2, 103-109 (1976) · Zbl 0319.60008
[27] Ismail, Mourad E. H., Complete monotonicity of modified Bessel functions, Proc. Amer. Math. Soc., 108, 2, 353-361 (1990) · Zbl 0685.33004 · doi:10.2307/2048282
[28] Ismail, Mourad E. H., Integral representations and complete monotonicity of various quotients of Bessel functions, Canad. J. Math., 29, 6, 1198-1207 (1977) · Zbl 0343.33005
[29] Miller, K. S.; Samko, S. G., Completely monotonic functions, Integral Transform. Spec. Funct., 12, 4, 389-402 (2001) · Zbl 1035.26012 · doi:10.1080/10652460108819360
[30] Baricz, {\'A}rp{\'a}d, Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc. (2), 53, 3, 575-599 (2010) · Zbl 1202.33008 · doi:10.1017/S0013091508001016
[31] Bordelon, D. J.; Ross, D. K., Problem 72-l5, “Inequalities for special functions”, SIAM Rev., 15, 665-670 (1973)
[32] RossSIAMREV151973 D. K. Ross, Problem 72-15, inequalities for special functions, SIAM Rev. 15 (1973), 668-670.
[33] Paris, R. B., An inequality for the Bessel function \(J_\nu (\nu x)\), SIAM J. Math. Anal., 15, 1, 203-205 (1984) · doi:10.1137/0515016
[34] Laforgia, Andrea, Bounds for modified Bessel functions, J. Comput. Appl. Math., 34, 3, 263-267 (1991) · Zbl 0726.33003 · doi:10.1016/0377-0427(91)90087-Z
[35] Weniger, Ernst Joachim; C\'{\i }zek, Ji{\v{r}}{\'{\i }}, Rational approximations for the modified Bessel function of the second kind, Comput. Phys. Comm., 59, 3, 471-493 (1990) · Zbl 0875.65036 · doi:10.1016/0010-4655(90)90089-J
[36] Luke, Yudell L., Inequalities for generalized hypergeometric functions, Collection of articles dedicated to J. L. Walsh on his 75th birthday, I, J. Approximation Theory, 5, 41-65 (1972) · Zbl 0225.33004
[37] Gaunt, Robert E., Inequalities for modified Bessel functions and their integrals, J. Math. Anal. Appl., 420, 1, 373-386 (2014) · Zbl 1297.26051 · doi:10.1016/j.jmaa.2014.05.083
[38] Kazarinoff, Donat K., On Wallis’ formula, Edinburgh Math. Notes, 1956, 40, 19-21 (1956) · Zbl 0072.28401
[39] Phillips, R. S.; Malin, Henry, Bessel function approximations, Amer. J. Math., 72, 407-418 (1950) · Zbl 0036.04001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.