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

Let K_{\nu } be the modified Bessel functions of the second kind of order \nu. The ratio 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 \left (-1\right) ^{n}Q_{\nu }^{\left (n\right) }\left (x\right) >\left (<\right) 0 for x>0 and n=2,3 if \left \vert \nu \right \vert >\left (<\right) 1/2 holds for n=2. This yields several new consequences and improves some known results. Finally, two conjectures are proposed. [ABSTRACT FROM AUTHOR]