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

Mao, Zhong-Xuan; Tian, Jing-Feng

Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 217-235

In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $-T_{\nu ,\alpha ,\beta }\left(s\right)$ is completely monotonic in $s$ and absolutely monotonic in $\nu$ if and only if $\beta \ge 1$, where $T_{\nu ,\alpha ,\beta }\left(s\right)=K_{\nu }^2\left(s\right)-\beta K_{\nu -\alpha }\left(s\right)K_{\nu +\alpha }\left(s\right)$ defined on $s>0$ and $K_{\nu }\left(s\right)$ is the modified Bessel function of the second kind of order $\nu$. Finally, we determine the necessary and sufficient conditions for the functions $s\mapsto T_{\mu ,\alpha ,1}\left(s\right)/T_{\nu ,\alpha ,1}\left(s\right)$, $s\mapsto (T_{\mu ,\alpha ,1}\left(s\right)+T_{\nu ,\alpha ,1}\left(s\right))/\left(2T_{(\mu +\nu )/2,\alpha ,1}\left(s\right)\right)$, and $s\mapsto \frac{{\mathrm{d}}^{n_1}}{\mathrm{d}{\nu }^{n_1}}{T}_{\nu ,\alpha ,1}\left(s\right)/\frac{{\mathrm{d}}^{n_2}}{\mathrm{d}{\nu }^{n_2}}{T}_{\nu ,\alpha ,1}\left(s\right)$ to be monotonic in $s\in (0,\infty )$ by employing the monotonicity rules.

Classification : 33C10, 33B15, 26A48

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