26D07, 26D20, 26E60.

Given two positive real numbers and , let , , and denote their arithmetic mean, geometric mean, and identric mean, respectively. Also, let for . In this note we prove that for all positive real numbers if and only if , and that for all positive real numbers if and only if . 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]. ;