Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means

Abstract

We give several sharp bounds for the Neuman means $N_{AH}$ and $N_{HA}$ ($N_{CA}$ and $N_{AC}$) in terms of harmonic mean H (contraharmonic mean C) or the geometric convex combination of arithmetic mean A and harmonic mean H (contraharmonic mean C and arithmetic mean A) and present a new chain of inequalities for certain bivariate means.