Complete monotonicity of modified Bessel functions

Abstract:

We prove that if $\nu > 1/2$, then ${2^{\nu - 1}}\Gamma (\nu )/[{x^{\nu /2}}{e^{\sqrt x }}{K_\nu }(\sqrt x )]$ is the Laplace transform of a selfdecomposable probability distribution while ${2^\nu }\Gamma \left ( {\nu + 1} \right ){x^{ - \nu /2}}{e^{ - \sqrt x }}{I_\nu }\left ( {\sqrt x } \right )$ is the Laplace transform of an infinitely divisible distribution. The former result is used to show that an estimate of ${\text {M}}$. Wong [13] is sharp. We also prove that the roots of the equations \[ {b^3}{l_{\nu - 1}}\left ( {a\sqrt z } \right )/{I_\nu }\left ( {a\sqrt z } \right ) = {a^3}{I_{\nu - 1}}\left ( {b\sqrt z } \right )/{I_\nu }\left ( {b\sqrt z } \right ),\] and \[ {b^3}{K_{\nu + 1}}\left ( {a\sqrt z } \right )/{K_\nu }\left ( {a\sqrt z } \right ) = {a^3}{K_{\nu + 1}}\left ( {b\sqrt z } \right )/{K_\nu }\left ( {b\sqrt z } \right ),\nu > 0,z \ne 0,\] lie in a certain sector contained in the open left half plane. This proves and extends a conjecture of ${\text {H}}$. Hattori arising from his work in partial differential equations.