We prove that if ν > 1 / 2 \nu > 1/2 , then 2 ν − 1 Γ ( ν ) / [ x ν / 2 e x K ν ( x ) ] {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 ν Γ ( ν + 1 ) x − ν / 2 e − x I ν ( x ) {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 M {\text {M}} . Wong [13] is sharp. We also prove that the roots of the equations \[ b 3 l ν − 1 ( a z ) / I ν ( a z ) = a 3 I ν − 1 ( b z ) / I ν ( b z ) , {b^3}{l_{\nu - 1}}\left ( {a\sqrt z } \right )/{I_\nu }\left ( {a\sqrt z } \right ) = {a^3}{I_{\nu - 1}}\left ( {b\sqrt z } \right )/{I_\nu }\left ( {b\sqrt z } \right ), \] and \[ b 3 K ν + 1 ( a z ) / K ν ( a z ) = a 3 K ν + 1 ( b z ) / K ν ( b z ) , ν > 0 , z ≠ 0 , {b^3}{K_{\nu + 1}}\left ( {a\sqrt z } \right )/{K_\nu }\left ( {a\sqrt z } \right ) = {a^3}{K_{\nu + 1}}\left ( {b\sqrt z } \right )/{K_\nu }\left ( {b\sqrt z } \right ),\nu > 0,z \ne 0, \] lie in a certain sector contained in the open left half plane. This proves and extends a conjecture of H {\text {H}} . Hattori arising from his work in partial differential equations.