On the Monotonicity and Log-Convexity of a Four-Parameter Homogeneous Mean

Abstract

A four-parameter homogeneous mean 𝐅(p,q;r,s;a,b) is defined by another approach. The criterion of its monotonicity and logarithmically convexity is presented, and three refined chains of inequalities for two-parameter mean values are deduced which contain many new and classical inequalities for means.