The aim of this paper is to establish a new upper bound for the positive zeros contained in the following theorem: Let denote the -th positive zero of the derivative of Bessel functions for , and . Then , where
and is the -th positive zero of the derivative of the Airy function. The bound is sharp for large values of and improves known results. A similar inequality holds for the -th positive zero of , too, where denotes the Bessel function of the second kind.