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REDHEFFER TYPE INEQUALITY FOR BESSEL FUNCTIONS

In this short note, by using mathematical induction and infinite product representations of the functionsJp : R → (−∞, 1] andIp : 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 functionIp are established. Here Jp andIp, denotes the Bessel function and modified Bessel function, whileΓ stands for the Euler gamma function. At the end of this work a lower bound for theΓ 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.