Sharp bounds involving the Sandor-Yang means in terms of other bivariate means

Abstract

In this paper, we present the best possible parameters ${\alpha _1},{\alpha _2},{\alpha _3},{\alpha _4},{\beta _1},{\beta _2},{\beta _3},{\beta _4} \in [0,1]$ such that the double inequalities hold for all $a,b > 0$ with $a \neq b$. Here $G\left( {a,b} \right)$, $A\left( {a,b} \right)$ and $Q\left( {a,b} \right)$ denote respectively the classical geometric, arithmetic and quadratic means of $a$ and $b$, and ${R_{GA}}\left( {a,b} \right) = X\left( {a,b} \right)$, ${R_{AG}}\left( {a,b} \right) = I\left( {a,b}\right)$, ${R_{QA}}\left( {a,b} \right)$ and ${R_{AQ}}\left( {a,b} \right)$ are S\'{a}ndor, identric and two S\'{a}ndor -Yang means derived from the Schwab-Borchardt mean.