The aim of this work is to extend the results obtained by Yang Bicheng and L. Debnath in [\emph{Some inequalities involving the constant e and an application to Carleman's inequality}, J. Math. Anal. Appl., extbf{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.