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Estimates of (1+x)1/x involved in Carleman’s inequality and Keller’s limit

Mortici Cristinel (Valahia University of Târgovişte, Târgovişte, Romania + Academy of Romanian Scientists, Bucharest, Romania)
Jang X.J. (Tsinghua University, Department of Mathematics, Beijing, China)

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.

Keywords: constant e, Carleman’s inequality, Keller’s limit, approximations