Further mean inequalities via some integral inequalities

Abstract

Recently intensive efforts are deployed to establish inequalities involving some bivariate means. In this paper we aim to investigate further mean inequalities. The approach employed here is based on a combination of the integral representations of the involved means with some advanced integral inequalities known in the literature. In particular, applying the Gr?ss inequality and the Ostrowski inequality we derive a lot of estimations for differences between some standard means and weighted means.

On some inequalities involving Beta-Logarithmic function

“…BL(1, 1; x, y) = B(x, y) and BL(a, b; 1, 1) = L(a, b), (1.6) where L(a, b) is defined for two positive numbers as follows [14], Recently, an important study on means has been undertaken in [17] based on their representation by integrals. The fundamental target of the current work is to establish approximations for the beta-logarithmic function BL(a, b; x, y).…”