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

Abstract

Let I v (x) be modified Bessel functions of the first kind. We prove the monotonicity property of the function x I u (x) I v (x) /I (u+v)/2 (x) 2 on (0,) . As a direct consequence, it deduces some known results including Turn-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 (x) and their ratios are presented.