Functional inequalities involving Bessel and modified Bessel functions of the first kind (Q935413)

Since the sine and cosine functions and the hyperbolic sine and cosine functions are particular cases of Bessel and modifed Bessel functions, the author extends the inequality \(\cosh x<({{\sinh x}\over{x}})^{3}\), \(x\neq 0\), proved by I. Lazarevic to a function \(G_{p}(x)=2^{p}\Gamma (p+1)x^{-p}I_{p}(x)\), \(p>-1\), where \(\Gamma\) is Euler gamma function and \(I_{p}(x)\) is the modified Bessel function of first kind. Some properties and inequalities satisfied by \(G_{p}(x)\) are proved in order to improve some known results and to present some new inequalities involving Bessel functions of the first kind.