Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean

Abstract

We give the greatest values $r_{\mathrm{1}}$, $r_{\mathrm{2}}$ and the least values $s_{\mathrm{1}}$, $s_{\mathrm{2}}$ in (1/2, 1) such that the double inequalities and hold for any $\alpha \in (\mathrm{0,1})$ and all $a,b>\mathrm{0}$ with $a\ne b$, where $A(a,b)$, $M(a,b)$, $C(a,b),$ and $T(a,b)$ are the arithmetic, Neuman-Sándor, contraharmonic, and second Seiffert means of $a$ and $b$, respectively.