Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

Abstract

For $1<r<+\infty$, we find the least value $\alpha$ and the greatest value $\beta$ such that the inequality $H_{\alpha }(a,b)<A_r(a,b)<H_{\beta }(a,b)$ holds for all $a,b>0$ with $a\ne b$. Here, $H_{\omega }(a,b)$ and $A_r(a,b)$ are the generalized Heronian and the power means of two positive numbers $a$ and $b$, respectively.