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

Abstract

We give several sharp bounds for the Neuman means NAH and NHA (NCA and NAC) 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.