The monotonicity and convexity for the ratios of modified Bessel functions of the second kind and applications

Source abstract

Let K v ( x ) K_{v}\left ( x\right ) be the modified Bessel functions of the second kind of order v v . We prove that the function x ↦ K u ( x ) K v ( x ) / K ( u + v ) / 2 ( x ) 2 x\mapsto K_{u}\left ( x\right ) K_{v}\left ( x\right ) /K_{\left ( u+v\right ) /2}\left ( x\right ) ^{2} is strictly decreasing on ( 0 , ∞ ) \left ( 0,\infty \right ) . Our study not only involves the Turán type inequalities, log-convexity or log-concavity of K v ( x ) K_{v}\left ( x\right ) , and the conjecture posed by Baricz, but also yields various new results concerning the monotonicity and convexity of the ratios of the modified Bessel functions of the second kind. As applications of our main theorems, some new sharp inequalities involving K v ( x ) K_{v}\left ( x\right ) are presented, which contain sharp estimates for K v ( x ) K_{v}\left ( x\right ) and sharp bounds for the ratios K v ′ ( x ) / K v ( x ) K_{v}^{\prime }\left ( x\right ) /K_{v}\left ( x\right ) and K v + 1 ( x ) / K v ( x ) K_{v+1}\left ( x\right ) /K_{v}\left ( x\right ) .