On Some Properties of Dirichlet's Eta Function

Abstract

In this paper, we establish some inequalities and monotonicity properties involving the Dirichlet's eta function. Thus, we improve and complement some previous results of Cerone and Dragomir and that of Alzer and Kwong. We also provide some new results. The tools employed in proving our results include some inequalities for the Nielsen's beta function, the logarithmic concavity property of the Dirichlet's eta function, the Cauchy-Schwarz integral inequality, and the classical mean value theorem.