On the Means of Jordan’s Totient Function

In this paper, we study some important means of Jordan’s totient function, especially, we obtain asymptotic formula for geometric mean and harmonic mean. We also study alternating sums of Jordan’s totient function and Carleman’s inequality for this function.

On some convergences to the constant e and improvements of Carleman’s inequality

We present sharp inequalities related to the sequence (1 + 1/n)n and some applications to Kellers’ limit and Carleman’s inequality.

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

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.

Some refinements and generalizations of Carleman's inequality

We give some refinements and generalizations of Carleman's inequality with weaker condition for weight coefficient.