Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind
In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $-T_{\nu ,\alpha ,\beta }(s)$ is completely monotonic in $s$ and absolutely monotonic in $\nu $ if and only if $\beta \ge 1$, where $T_{\nu ,\alpha ,\beta }(s)=K_{\nu }^2(s)-...
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| Language: | English |
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.399/ |
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| author | Mao, Zhong-Xuan Tian, Jing-Feng |
| author_facet | Mao, Zhong-Xuan Tian, Jing-Feng |
| author_sort | Mao, Zhong-Xuan |
| collection | DOAJ |
| description | In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $-T_{\nu ,\alpha ,\beta }(s)$ is completely monotonic in $s$ and absolutely monotonic in $\nu $ if and only if $\beta \ge 1$, where $T_{\nu ,\alpha ,\beta }(s)=K_{\nu }^2(s)-\beta K_{\nu -\alpha }(s)K_{\nu +\alpha }(s)$ defined on $s>0$ and $K_{\nu }(s)$ is the modified Bessel function of the second kind of order $\nu $. Finally, we determine the necessary and sufficient conditions for the functions $s \mapsto T_{\mu ,\alpha ,1}(s)/T_{\nu ,\alpha ,1}(s)$, $s \mapsto (T_{\mu ,\alpha ,1}(s) + T_{\nu ,\alpha ,1}(s))/(2T_{(\mu +\nu )/2,\alpha ,1}(s))$, and $s \mapsto \frac{\mathrm{d}^{n_1}}{\mathrm{d} \nu ^{n_1}} T_{\nu ,\alpha ,1}(s)/\frac{\mathrm{d}^{n_2}}{\mathrm{d} \nu ^{n_2}} T_{\nu ,\alpha ,1}(s)$ to be monotonic in $s\in (0,\infty )$ by employing the monotonicity rules. |
| format | Article |
| id | doaj-art-a34cdd4cfece4495a7337ad74d51ffd5 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-a34cdd4cfece4495a7337ad74d51ffd52025-02-07T11:06:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G121723510.5802/crmath.39910.5802/crmath.399Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kindMao, Zhong-Xuan0Tian, Jing-Feng1Department of Mathematics and Physics, North China Electric Power University,Yonghua Street 619, 071003, Baoding, P. R. ChinaDepartment of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, P. R. ChinaIn this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $-T_{\nu ,\alpha ,\beta }(s)$ is completely monotonic in $s$ and absolutely monotonic in $\nu $ if and only if $\beta \ge 1$, where $T_{\nu ,\alpha ,\beta }(s)=K_{\nu }^2(s)-\beta K_{\nu -\alpha }(s)K_{\nu +\alpha }(s)$ defined on $s>0$ and $K_{\nu }(s)$ is the modified Bessel function of the second kind of order $\nu $. Finally, we determine the necessary and sufficient conditions for the functions $s \mapsto T_{\mu ,\alpha ,1}(s)/T_{\nu ,\alpha ,1}(s)$, $s \mapsto (T_{\mu ,\alpha ,1}(s) + T_{\nu ,\alpha ,1}(s))/(2T_{(\mu +\nu )/2,\alpha ,1}(s))$, and $s \mapsto \frac{\mathrm{d}^{n_1}}{\mathrm{d} \nu ^{n_1}} T_{\nu ,\alpha ,1}(s)/\frac{\mathrm{d}^{n_2}}{\mathrm{d} \nu ^{n_2}} T_{\nu ,\alpha ,1}(s)$ to be monotonic in $s\in (0,\infty )$ by employing the monotonicity rules.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.399/ |
| spellingShingle | Mao, Zhong-Xuan Tian, Jing-Feng Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind Comptes Rendus. Mathématique |
| title | Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind |
| title_full | Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind |
| title_fullStr | Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind |
| title_full_unstemmed | Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind |
| title_short | Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind |
| title_sort | monotonicity and complete monotonicity of some functions involving the modified bessel functions of the second kind |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.399/ |
| work_keys_str_mv | AT maozhongxuan monotonicityandcompletemonotonicityofsomefunctionsinvolvingthemodifiedbesselfunctionsofthesecondkind AT tianjingfeng monotonicityandcompletemonotonicityofsomefunctionsinvolvingthemodifiedbesselfunctionsofthesecondkind |