The extremal values of the ratio of differences of power mean, arithmetic mean, and geometric mean

Abstract: In the paper the maximum and the minimum of the ratio of the difference of the arithmetic mean and the geometric mean, and the difference of the power mean and the geometric mean of $n$ variables, are studied. A new optimization argument was used which reduces $n$ variable optimization problem to a single variable. All possible cases of the choice of the power mean and the choice of the number of variables of the means are studied. The obtained results generalize and complete the earlier results which were either for specific intervals of power means or for small number of variables of the means. Some of the results are formulated as the best constant inequalities involving interpolation of the arithmetic mean and the geometric mean. The monotonicity and convergence of these best constants are also studied.