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].
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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 
. 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].
