ON SOME PROPERTIES OF HYPER-BESSEL an‎d RELATED FUNCTIONS

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 7→ Jαd (x) on some sets. Also, we show that the function x 7→ Jαd (x) is a decreasing function on [0, jαd,1), and the function x 7→ xI 0 αd ( d+1√ x) Iαd ( 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.