A note on monotonicity property of Bessel functions

Abstract

A theorem of Lorch, Muldoon and Szegö states that the sequence {∫jα,kjα,k+1t−α|Jα(t)|dt}k=1∞ is decreasing for α>−1/2, where Jα(t) the Bessel function of the first kind order α and jα,k its kth positive root. This monotonicity property implies Szegö's inequality ∫0xt−αJα(t)dt≥0, when α≥α′ and α′ is the unique solution of ∫0jα,2t−αJα(t)dt=0.We give a new and simpler proof of these classical results by expressing the above Bessel function integral as an integral involving elementary functions.