Convexity of a ratio of the modified Bessel functions of the second kind with applications

Let K ν K_{\nu } be the modified Bessel functions of the second kind of order ν \nu . The ratio Q ν ( x ) = x K ν − 1 ( x ) / K ν ( x ) Q_{\nu }\left ( x\right ) =xK_{\nu -1}\left ( x\right ) /K_{\nu }\left ( x\right ) appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that ( − 1 ) n Q ν ( n ) ( x ) > ( > ) 0 \left ( -1\right ) ^{n}Q_{\nu }^{\left ( n\right ) }\left ( x\right ) >\left ( >\right ) 0 for x > 0 x>0 and n = 2 , 3 n=2,3 if | ν | > ( > ) 1 / 2 \left \vert \nu \right \vert >\left ( >\right ) 1/2 holds for n = 2 n=2 . This yields several new consequences and improves some known results. Finally, two conjectures are proposed.