The following results are given simple proofs using a lemma that states that if satisfy the usual conditions for the mean-value theorem and if is increasing so are the ratios .
If then: ;
if then: ;
if then: .
are the arithmetic, geometric, logarithmic and identric means of two variables respectively. These results extend earlier inequalities of Trif, Sandor and Trif, Alzer and Qui, Zhu and Wu.