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.