Complete monotonicity of modified Bessel functions

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

$${b^3}{l_{\nu - 1}}\left( {a\sqrt z } \right)/{I_\nu }\left( {a\sq... ...a^3}{I_{\nu - 1}}\left( {b\sqrt z } \right)/{I_\nu }\left( {b\sqrt z } \right),$$

and

$${b^3}{K_{\nu + 1}}\left( {a\sqrt z } \right)/{K_\nu }\left( {a\sq... ...}\left( {b\sqrt z } \right)/{K_\nu }\left( {b\sqrt z } \right),\nu > 0,z \ne 0,$$

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.