| 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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