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Some new inequalities for means in two variables. (English) Zbl 1154.26029
The following results are given simple proofs using a lemma that states that if $f,g$ satisfy the usual conditions for the mean-value theorem and if $f'/g'$ is increasing so are the ratios $\bigl(f(x)-f(b)\bigr)/\bigl(g(x)-g(b)\bigr), \bigl(f(x)-f(a)\bigr)/\bigl(g(x)-g(a)\bigr)$. If $p\ge 1$ then: $\alpha_pA^p+(1- \alpha_p)G^p<L^p< \beta_pA^p+(1- \beta_p)G^p \Leftrightarrow \alpha_p\le 0 \land\beta_p\ge 1/3$; if $0\le p\le 6/5$ then: $\alpha_pA^p+(1- \alpha_p)G^p<I^p< \beta_pA^p+(1- \beta_p)G^p \Leftrightarrow \alpha_p\le 2/3 \land\beta_p\ge (2/e)^p$; if $p\ge 2$ then: $\alpha_pA^p+(1- \alpha_p)G^p<I^p< \beta_pA^p+(1- \beta_p)G^p \Leftrightarrow \alpha_p\le (2/e)^p \land\beta_p\ge 2/3$. $A,G, L, I$ 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.

26D07Inequalities involving other types of real functions