Grünbaum’s Inequality for Bessel Functions and Its Extensions

Abstract

Recently, F. A. Grunbaum found a new kind of inequality for the Bessel functions, namely $1 + \mathcal{J}_\nu (a) \geqq \mathcal{J}_\nu (b) + \mathcal{J}_\nu (c)$$(a^2 = b^2 + c^2 ,\nu \geqq 0,\mathcal{J}_\nu (x) = \Gamma (\nu + 1)({2 / x})^\nu J_\nu (x))$. Later, another proof was given by R. Askey. In his paper Grunbaum suggested the desirability of finding a proof which, when $\nu = {n / {2 - 1}}$, $n = 2,3, \cdots $, made use of the property of these functions of being spherical functions for the corresponding symmetric spaces. Such a proof is given in the present note, and it is found that the method provides an extension of Grunbaum’s inequality as well as other inequalities of a similar nature.