A note on monotonicity property of Bessel functions

摘要

A theorem of Lorch, Muldoon and Szeg? states that the sequence{∫jα,kjα,k+1t?α|Jα(t)|dt}k=1∞is decreasing forα>?1/2, whereJα(t)the Bessel function of the first kind orderαandjα,kitskthpositive root. This monotonicity property implies Szeg?'s inequality∫0xt?αJα(t)dt≥0,whenα≥α′andα′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 functionintegral as an integral involving elementary functions.