Redheffer Type Inequality for Bessel Functions

In this short note, by using mathematical induction and infinite product representations of the functions Jp : R → (−∞, 1] and Ip : R → [1,∞), defined by Jp(x) = 2Γ(p + 1)xJp(x) and Ip(x) = 2Γ(p + 1)xIp(x), an extension of Redheffer’s inequality for the function Jp and a Redheffer-type inequality for the function Ip are established. Here Jp and Ip, denotes the Bessel function and modified Bessel function, while Γ stands for the Euler gamma function. At the end of this work a lower bound for the Γ function is deduced, using Euler’s infinite product formula. Our main motivation to write this note is the publication of C.P. Chen, J.W. Zhao and F. Qi [2], which we wish to complement.