We give several sharp bounds for the Neuman means and ( and ) 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.
Zhi-Jun Guo. Yu-Ming Chu. Ying-Qing Song. Xiao-Jing Tao. "Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means." Abstr. Appl. Anal. 2014 1 - 8, 2014. https://doi.org/10.1155/2014/914242