On some properties of hyper-bessel and related functions

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