ON SOME PROPERTIES OF HYPER-BESSEL AND RELATED FUNCTIONS

I. AKTAS

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Abstract

In this study, by using the Hadamard product representation of the hyper- Bessel function and basic arithmetic operations in mathematics we investigate the sign of the hyper-Bessel function x 7! Jd (x) on some sets. Also, we show that the function x 7! Jd (x) is a decreasing function on [0; jd;1), and the function x 7! xI0 d ( d+1 p x) Id ( d+1 p x) is an increasing function on (0;1), where jd;1 and Id denote the rst positive zero of the function Jd (x) and modied hyper-Bessel function, respectively. In addition, we prove the strictly log-concavity of the functions Jd (x) and Jd (x) on some sets. Moreover, we give some illustrative examples regarding our main results.

Keywords

Decreasing and increasing functions, Hadamard product representation, hyper- Bessel function, log-concavity, modied hyper-Bessel function.