Some Applications related to the Arithmetic-Geometric Mean Inequality

Abstract

In this paper, we introduce some refinements of the arithmetic-geometric-logarithmic mean inequality. The obtained results allow us to get new inequalities for the inverse sine function and refined inequalities for well-known hyperbolic inequalities.  Further, as a new track in this field, we present a possible arithmetic-geometric mean inequality in the complex plane, with applications towards complex inequalities for concave functions and their sub-additive behavior. Moreover, the celebrated Jensen-Mercer inequality for convex functions is refined in a way that implies some new relations among the arithmetic, geometric, Heinz, and weighted logarithmic means.