New Bounds for the Identric Mean of Two Arguments

Given two positive real numbers x and y, let A(x, y), G(x, y), and I(x, y) denote their arithmetic mean, geometric mean, and identric mean, respectively. Also, let Kp(x, y) = p √ 2 3A p(x, y) + 13G p(x, y) for p > 0. In this note we prove that Kp(x, y) < I(x, y) for all positive real numbers x 6= y if and only if p ≤ 6/5, and that I(x, y) < Kp(x, y) for all positive real numbers x 6= y if and only if p ≥ (ln 3− ln 2)/(1− ln 2). These results, complement and extend similar inequalities due to J. Sándor [2], J. Sándor and T. Trif [3], and H. Alzer and S.-L. Qiu [1].