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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1778-3569 |