@article {
author = {Ito, Masatoshi},
title = {Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator inequalities},
journal = {Advances in Operator Theory},
volume = {3},
number = {4},
pages = {763-780},
year = {2018},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.15352/aot-1801-1303},
abstract = {As generalizations of the arithmetic and the geometric means, for positive real numbers $a$ and $b$,the power difference mean $J_{q}(a,b)=\frac{q}{q+1}\frac{a^{q+1}-b^{q+1}}{a^{q}-b^{q}}$,the Lehmer mean $L_{q}(a,b)=\frac{a^{q+1}+b^{q+1}}{a^{q}+b^{q}}$ and the Heron mean $K_{q}(a,b)=(1-q)\sqrt{ab}+q\frac{a+b}{2}$ are well known.In this paper, concerning our recent results on estimations of the power difference mean, we obtain the greatest value $\alpha=\alpha(q)$ and the least value $\beta=\beta(q)$ such that the double inequalityfor the Lehmer mean \[K_{\alpha}(a,b)