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Inequalities between arithmetic-geometric, Gini, and Toader means. (English) Zbl 1229.26031
Summary: We find the greatest values $p_1, p_2$ and least values $q_1, q_2$ such that the double inequalities $S_{p_1}(a, b) < M(a, b) < S_{q_1}(a, b)$ and $S_{p_2}(a, b) < T(a, b) < S_{q_2}(a, b)$ hold for all $a, b > 0$ with $a \neq b$ and present some new bounds for the complete elliptic integrals. Here $M(a, b), T(a, b)$, and $S_p(a, b)$ are the arithmetic-geometric, Toader, and $p$th Gini means of two positive numbers $a$ and $b$, respectively.

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
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