Inequalities on an extended Bessel function

This paper studies an extended Bessel function of the form [Formula: see text] Representation formulations for [Formula: see text] are derived in terms of the parameters a, b, and p. An important consequence is the derivation of an [Formula: see text]-order differential equation satisfied by the function [Formula: see text]. Interesting functional inequalities are established, particularly for the case [Formula: see text], and [Formula: see text]. Monotonicity properties of [Formula: see text] are also studied for non-positive c. Log-concavity and log-convexity properties in terms of the parameters d and p are respectively investigated for the closely related function [Formula: see text] which leads to direct and reverse Turán-type inequalities.