Comparison of differences between arithmetic and geometric means

Abstract

We complement a recent result of S. Furuichi, by showing that the differences ${\sum }_{i=1}^n{\alpha }_i{x}_i-{\prod }_{i=1}^n{x}_i^{{\alpha }_i}$ associated to distinct sequences of weights are comparable, with constants that depend on the smallest and largest quotients of the weights.

Improvements on the bounds after becoming aware of the paper by Sever S. Dragomir, "Bounds for the normalised Jensen functional", Bull. Austral. Math. Soc. 74 (2006), no. 3, 471--478. Correction of an error and of some typos