Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications

Abstract

Let $I_{v}\left( x\right) $ be modified Bessel functions of the first kind. We prove the monotonicity property of the function $x\mapsto I_{u}\left( x\right) I_{v}\left( x\right) /I_{\left( u+v\right) /2}\left( x\right) ^{2}$ on $\left( 0,\infty \right) $. As a direct consequence, it deduces some known results including Tur\'{a}n-type inequalities and log-convexity or log-concavity of $I_{v}$ in $v$, as well as it yields some new and interesting monotonicity and convexity concerning the ratios of modified Bessel functions of the first kind. In addition, a few of sharp bounds involving $I_{v}\left( x\right) $ and their ratios are presented.