On some properties of hyper-Bessel and related functions

Abstract

In this study, by using the Hadamard product representation of the hyperBessel function and basic concepts in mathematics we investigate the sign of the hyperBessel function x → Jαd (x) on some sets. Also, we show that the function x → Jαd (x) is a decreasing function on [0, jαd,1), and the function x → xII′αd ( d+1√ x) / II’αd ( d+1√ x) is an increasing function on (0, ∞), where jαd,1 and Iαd denote the first positive zero of the function Jαd (x) and modified hyper-Bessel function, respectively. In addition, we prove the strictly log-concavity of the functions Jαd (x) and Jαd (x) on some sets. Moreover, we give some illustrative examples regarding our main results.