Communications of the
Korean Mathematical Society CKMS

This article studies the monotonicity, log-convexity of the modified Lommel functions by using its power series and infinite product representation. Some properties for the ratio of the modified Lommel functions with the Lommel function, $\sinh$ and $\cosh$ are also discussed. As a consequence, 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.

Keywords: Lommel functions, modified Lommel functions, Tur\'an-type inequality, monotonicity properties, log-convexity

MSC numbers: 33C10, 26D7, 26D15