Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

Abstract

We find the greatest value $\alpha$ and the least value $\beta$ in $\mathrm{(1}/2,\mathrm{1)}$ such that the double inequality holds for all $a,b>0$ with $a\ne b$. Here, $T(a,b)=(a-b)/[$2 arctan$\left(\right(a-b)/(a+b\left)\right)]$ and $C\left(a,b\right)=(a^2+b^2)/(a+b)$ are the Seiffert and contraharmonic means of $a$ and $b$, respectively.