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

Abstract

Abstract In this paper, we find the greatest values α 1 , α 2 $\alpha_{1},\alpha_{2}$ and the smallest values β 1 , β 2 $\beta_{1},\beta_{2}$ such that the double inequalities L α 1 ( a , b ) 0 $a, b>0$ with a ≠ b $a\neq b$ , where AG ( a , b ) $\operatorname{AG}(a,b)$ , T ( a , b ) $T(a,b)$ and L p ( a , b ) $L_{p}(a,b)$ are the arithmetic-geometric, Toader and generalized logarithmic means of two positive numbers a and b, respectively.

Keywords

• arithmetic-geometric mean
• Toader mean
• logarithmic mean