Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

Zhong-Xuan Mao; Jing-Feng Tian

doi:10.5802/crmath.399

Equations aux dérivées partielles

Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

Zhong-Xuan Mao ¹ ; Jing-Feng Tian ²

¹ Department of Mathematics and Physics, North China Electric Power University,Yonghua Street 619, 071003, Baoding, P. R. China
² Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, P. R. China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 217-235.

Résumé

In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $- T_{ν, α, β} (s)$ is completely monotonic in $s$ and absolutely monotonic in $ν$ if and only if $β \geq 1$ , where $T_{ν, α, β} (s) = K_{ν}^{2} (s) - β K_{ν - α} (s) K_{ν + α} (s)$ defined on $s > 0$ and $K_{ν} (s)$ is the modified Bessel function of the second kind of order $ν$ . Finally, we determine the necessary and sufficient conditions for the functions $s \mapsto T_{μ, α, 1} (s) / T_{ν, α, 1} (s)$ , $s \mapsto (T_{μ, α, 1} (s) + T_{ν, α, 1} (s)) / (2 T_{(μ + ν) / 2, α, 1} (s))$ , and $s \mapsto \frac{d^{n_{1}}}{d ν^{n_{1}}} T_{ν, α, 1} (s) / \frac{d^{n_{2}}}{d ν^{n_{2}}} T_{ν, α, 1} (s)$ to be monotonic in $s \in (0, \infty)$ by employing the monotonicity rules.

Reçu le : 2022-03-14
Accepté le : 2022-06-18
Publié le : 2023-01-12

DOI : 10.5802/crmath.399

Classification : 33C10, 33B15, 26A48

Affiliations des auteurs :

Zhong-Xuan Mao ¹ ; Jing-Feng Tian ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G1_217_0,
     author = {Zhong-Xuan Mao and Jing-Feng Tian},
     title = {Monotonicity and complete monotonicity of some functions involving the modified {Bessel} functions of the second kind},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {217--235},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.399},
     language = {en},
}

TY  - JOUR
AU  - Zhong-Xuan Mao
AU  - Jing-Feng Tian
TI  - Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 217
EP  - 235
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.399
LA  - en
ID  - CRMATH_2023__361_G1_217_0
ER  -

%0 Journal Article
%A Zhong-Xuan Mao
%A Jing-Feng Tian
%T Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind
%J Comptes Rendus. Mathématique
%D 2023
%P 217-235
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.399
%G en
%F CRMATH_2023__361_G1_217_0

Zhong-Xuan Mao; Jing-Feng Tian. Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 217-235. doi : 10.5802/crmath.399. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.399/

Bibliographie
Cité par

[1] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Milton Abramowitz; I. A. Stegun, eds.), Dover Publications, 1964 | DOI | Zbl

[2] Árpád Baricz Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc., II. Ser., Volume 53 (2010) no. 3, pp. 575-599 | DOI | MR | Zbl

[3] Árpád Baricz Turán type inequalities for modified Bessel functions, Bull. Aust. Math. Soc., Volume 82 (2010) no. 2, pp. 254-264 | DOI | Zbl

[4] Árpád Baricz Bounds for Turánians of modified Bessel functions, Expo. Math., Volume 33 (2015) no. 2, pp. 223-251 | DOI | Zbl

[5] Árpád Baricz; Mourad E. H. Ismail Turán type inequalities for Tricomi confluent hypergeometric functions, Constr. Approx., Volume 37 (2013) no. 2, pp. 195-221 | DOI | Zbl

[6] Árpád Baricz; Dragana Jankov; Tibor K. Pogány Turán type inequalities for Krätzel functions, J. Math. Anal. Appl., Volume 388 (2012) no. 2, pp. 716-724 | DOI | Zbl

[7] Árpád Baricz; Tibor K. Pogány Turán determinants of Bessel functions, Forum Math., Volume 26 (2014) no. 1, pp. 295-322 | Zbl

[8] Árpád Baricz; Saminathan Ponnusamy On Turán type inequalities for modified Bessel functions, Proc. Am. Math. Soc., Volume 141 (2013) no. 7, pp. 523-532 | Zbl

[9] Árpád Baricz; Saminathan Ponnusamy; Sanjeev Singh Turán type inequalities for Struve functions, J. Math. Anal. Appl., Volume 445 (2017) no. 1, pp. 971-984 | DOI | Zbl

[10] S. Bernstein Sur les fonctions absolument monotones(French), Acta Math., Volume 52 (1928), pp. 1-66 | DOI | Zbl

[11] Mieczysław Biernacki; Jan Krzyż On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska, Sect. A, Volume 9 (1957), pp. 135-147 | Zbl

[12] Martin Bohner; Tom Cuchta The Bessel difference equation, Proc. Am. Math. Soc., Volume 145 (2017) no. 4, pp. 1567-1580 | DOI | MR | Zbl

[13] Jeff Cheeger; Michael Gromov Mikhail Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., Volume 17 (1982), pp. 15-53 | MR | Zbl

[14] Luc Devroye Simulating Bessel random variables, Stat. Probab. Lett., Volume 57 (2002) no. 3, pp. 249-257 | DOI | MR | Zbl

[15] Robert E. Gaunt Inequalities for some integrals involving modified Bessel functions, Proc. Am. Math. Soc., Volume 147 (2019) no. 7, pp. 2937-2951 | DOI | MR | Zbl

[16] Emil Grosswald The student t-distribution of any degrees of freedom is infinitely divisible, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 36 (1976), pp. 103-109 | DOI | MR | Zbl

[17] Mourad E. H. Ismail Bessel functions and the infinite divisibility of the student $t$ -distribution, Ann. Probab., Volume 5 (1977), pp. 582-585 | MR | Zbl

[18] Mourad E. H. Ismail; Martin E. Muldoon Monotonicity of the zeros of a cross-product of Bessel functions, SIAM J. Math. Anal., Volume 9 (1978), pp. 759-767 | DOI | MR | Zbl

[19] Feng Qi Decreasing properties of two ratios defined by three and four polygamma functions, C. R. Math. Acad. Sci. Paris, Volume 360 (2022), pp. 89-101 | MR | Zbl

[20] Christian Robert Modified Bessel functions and their applications in probability and statistics, Stat. Probab. Lett., Volume 9 (1990) no. 2, pp. 155-161 | DOI | MR | Zbl

[21] Javier Segura Bounds for ratios of modified Bessel functions and associated Turán-type inequalities, J. Math. Anal. Appl., Volume 374 (2011) no. 2, pp. 516-528 | DOI | Zbl

[22] Shan Tan; Licheng Jiao Multishrinkage: Analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model, Optics Lett., Volume 32 (2007) no. 17, pp. 2583-2585 | DOI

[23] Pál Turán On the zeros of the polynomials of Legendre, Čas. Pěst. Mat. Fys., Volume 75 (1950), pp. 113-122 | DOI | MR | Zbl

[24] Henk Van Haeringen Bound states for r $^{- 2}$ -like potentials in one and three dimension, J. Math. Phys., Volume 19 (1978) no. 10, pp. 2171-2179 | DOI | MR

[25] George N. Watson A treatise on the theory of bessel functions, Cambridge University Press, 1944 | Zbl

[26] David V. Widder The Laplace Transform, Princeton Mathematical Series, 6, Princeton University Press, 1941 | MR | Zbl

[27] Zhen-Hang Yang; Yu-Ming Chu On approximating the modified Bessel function of the second kind, J. Inequal. Appl., Volume 2017 (2017), 41, 8 pages | MR | Zbl

[28] Zhen-Hang Yang; Yu-Ming Chu Monotonicity and inequalities involving the modified Bessel functions of the second kind, J. Math. Anal. Appl., Volume 508 (2022) no. 2, 125889, 23 pages | MR | Zbl

[29] Zhen-Hang Yang; Yu-Ming Chu; Miao-Kun Wang Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl., Volume 428 (2015) no. 1, pp. 587-604 | DOI | MR | Zbl

[30] Zhen-Hang Yang; Shen-Zhou Zheng The monotonicity and convexity for the ratios of modified Bessel functions of the second kind and applications, Proc. Am. Math. Soc., Volume 145 (2017) no. 7, pp. 2943-2958 | DOI | MR | Zbl

[31] Tie-Hong Zhao; Lei Shi; Yu-Ming Chu Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM, Volume 114 (2020) no. 2, 96, 14 pages | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

From Boussinesq–Love contact to impact between hyperelastic bodies

Zhi-Qiang Feng; Claude Vallée

C. R. Méca (2007)

Fractional calculus in one-dimensional isotropic thermo-viscoelasticity

Magdy A. Ezzat; Ahmed S. El-Karamany; Alaa A. El-Bary; ...

C. R. Méca (2013)

Non-symmetric localized fold of a floating sheet

Marco Rivetti

C. R. Méca (2013)