Estimates of (1+x)^{1/x} Involved in Carleman's Inequality and Keller's Limit

Abstract

The aim of this work is to extend the results obtained by Yang Bicheng and L. Debnath in [Some inequalities involving the constant e and an application to Carleman's inequality, J. Math. Anal. Appl., 223 (1998), 347-353]. We present a simple proof of our new result which can be also used as a direct proof for Yang Bicheng and L. Debnath results. Finally some applications to generalized Keller's limit and further directions are provided.