  This article studies the monotonicity, log-convexity of the modified Lommel
functions by using its power series and infinite product representation. Same
properties for the ratio of the modified Lommel functions with the Lommel
function, $\sinh$ and $\cosh$ are also discussed. As a consequence, some
Tur\'an type and reverse Tur\'an type inequalities are given. A Rayleigh type
function for the Lommel functions are derived and as an application, we obtain
the Redheffer-type inequality.
