High order monotonicity of a ratio of the modified Bessel function with applications

Abstract

Let $K_{\mathcal{\nu }}$ be the modified Bessel functions of the second kind of order $\mathcal{\nu }$ and $Q_{\nu }\left( x\right) =xK_{\mathcal{\nu -}1}\left( x\right) /K_{\mathcal{\nu }}\left( x\right) $. In this paper, we proved that $Q_{\mathcal{\nu }}^{\prime \prime \prime }\left( x\right) <\left( >\right) 0$ for $x>0$ if $\left\vert \nu \right\vert >\left( <\right) 1/2$, which gives an affirmative answer to a guess. As applications, some monotonicity and concavity or convexity results as well functional inequalities involving $Q_{\nu }\left( x\right) $ are established. Moreover, several high order monotonicity of $x^{k}Q_{\nu }^{\left( n\right) }\left( x\right) $ on $\left( 0,\infty \right) $ for certain integers $k$ and $n$ are given.

Keywords

• modified Bessel function of the second kind
• high order monotonicity
• complete monotonicity
• functional inequality

References

1. [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Dover Publications, New York andWashington, 1972.
2. [2] M. D. Alenxandrov and A. A. Lacis, A new three-parameter cloud/aerosol particle size distribution based on the generalized inverse Gaussian density function, Appl. Math. Comput. 116, 153–165, 2000.
3. [3] A. Baricz, Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc. 53, 575–599, 2010.
4. [4] Á. Baricz, Bounds for Turánians of modified Bessel functions, Expo. Math. 2015 (2), 223–251, 2015.
5. [5] E. Grosswald, The Student t-distribution of any degree of freedom is infinitely divisible, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (2), 103–109, 1976.
6. [6] D. H. Kelker, Infinite divisibility and variance mixtures of the normal distribution, Ann. Math. Statist. 42, 802–808, 1971.
7. [7] M. E. H. Ismail, Bessel functions and the infinite divisibility of the Student tdistribution, Ann. Probability 5 (4), 582–585, 1977.
8. [8] M. E. H. Ismail, Intergal representations and complete monotonicity of various quotients of Bessel functions, Canadian J. Math. 29 (6), 1198–1207, 1977.

9. [9] M. E. H. Ismail and D. H. Kelker, The Bessel polynomials and the student tdistribution, SIAM J. Math. Anal. 7, 82–91, 1976.
10. [10] M. E. H. Ismail and M. E. Muldoon, Monotonicity of the zeros of a cross-product of Bessel functions, SIAM J. Math. Anal. 9 (4), 759–767, 1978.
11. [11] Z.-X. Mao and J.-F. Tian, Monotonicity and complete monotonicity of some functions involving the modified Bessel function of the second kind, C. R. Math. Acad. Sci. Paris 361, 217–235, 2023.
12. [12] M. Petrovic, Sur une équation fonctionnelle, Publ. Math. Univ. Belgrade 1, 149–156, 1932.
13. [13] R. A. Rosenbaum, Subadditive functions, Duke Math. J. 17 (1950), 227–242.
14. [14] J. Segura, Bounds for ratios of modified Bessel functions and associated Turán-type inequalities, J. Math. Anal. Appl. 374 (2), 516–528, 2011.
15. [15] H. C. Simpson and S. J. Spector, Some monotonicity results for ratios of modified Bessel functions, Quart. Appl. Math. 42 (1), 95–98, 1984.
16. [16] L. Trlifaj, Asymptotic ratios of Bessel functions of purely imaginary argument, Apl. Mat. 19, 1–5, 1974.
17. [17] H. Van Haeringen, Bound states for r-2-like potentials in one and three dimensions, J. Math. Phys. 19, 2171–2179, 1978.
18. [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944.
19. [19] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
20. [20] Z. Yang and J.-F. Tian, Monotonicity rules for the ratio of two Laplace transforms with applications, J. Math. Anal. Appl. 470 (2), 821–845, 2019.
21. [21] Z.-H. Yang and J.-F. Tian, A new chain of inequalities involving the Toader-Qi, logarithmic and exponential means, Appl. Anal. Discrete Math. 15 (2), 467–485, 2021.
22. [22] Z.-H. Yang and J.-F. Tian, Convexity of a ratio of the modified Bessel functions of the second kind with applications, Proc. Amer. Math. Soc. 150 (7), 2997–3009, 2022.
23. [23] Z.-H. Yang and J.-F. Tian, Convexity of ratios of the modified Bessel functions of the first kind with applications, Rev. Mat. Complut. 36 (3), 799–825, 2023.
24. [24] Z.-H. Yang and J.-F. Tian, The signs rule for the Laplace integrals with applications, Indian J. Pure Appl. Math. https://doi.org/10.1007/s13226-023-00447-6.
25. [25] Z.-H. Yang, J.-F. Tian and M.-K. Wang, A positive answer to Bhatia–Li conjecture on the monotonicity for a new mean in its parameter, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (3), Paper No. 126, 2020.
26. [26] Z.-H. Yang, J.-F. Tian and Y.-R. Zhu, A sharp lower bound for the complete elliptic integrals of the first kind, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (1), Paper No. 8, 17 pages, 2021.
27. [27] Z.-H. Yang and S.-Z. Zheng, The monotonicity and convexity for the ratios of modified Bessel functions of the second kind and applications, Proc. Amer. Math. Soc. 145, 2943-2958, 2017.
28. [28] Z.-H. Yang and S.-Z. Zheng, Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications, Math. Inequal. Appl. 21 (1), 107–125, 2018.