Inequalities Involving Bessel and Hypergeometric Functions

Abstract

In this chapter we deduce several functional inequalities for generalized Bessel functions of the first kind, Gaussian and Kummer hypergeometric functions as well as for general power series with positive coefficients. We present extensions to Bessel functions of some known trigonometric inequalities like Jordan, Cusa, van der Corput, Redheffer, Mahajan, Mitrinović. Moreover, we establish some Grünbaum, Askey and Landen type inequalities for generalized Bessel functions. The methods used to derive these inequalities are based on classical analysis. Among others, we use a criterion for the monotonicity of the quotient of two MacLaurin series and the monotone form of l’Hospital’s rule.