Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean.

In this paper, we find the greatest values $\alpha_{1},\alpha_{2}$ and the smallest values $\beta_{1},\beta_{2}$ such that the double inequalities $L_{\alpha_{1}}(a,b)<\operatorname{AG}(a,b)<L_{\beta_{1}}(a,b)$ and $L_{\alpha_{2}}(a,b)< T(a,b)< L_{\beta_{2}}(a,b)$ hold for all $a, b>0$ with $a\neq b$ , where $\operatorname{AG}(a,b)$ , $T(a,b)$ and $L_{p}(a,b)$ are the arithmetic-geometric, Toader and generalized logarithmic means of two positive numbers a and b, respectively. [ABSTRACT FROM AUTHOR]