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Zbl 1154.26029
Zhu, Ling
Some new inequalities for means in two variables. (English)
[J] Math. Inequal. Appl. 11, No. 3, 443-448 (2008). ISSN 1331-4343

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 $\bigl(f(x)-f(b)\bigr)/\bigl(g(x)-g(b)\bigr), \bigl(f(x)-f(a)\bigr)/\bigl(g(x)-g(a)\bigr)$. \par If $p\ge 1$ then: $\alpha_pA^p+(1- \alpha_p)G^p<L^p< \beta_pA^p+(1- \beta_p)G^p \Leftrightarrow \alpha_p\le 0 \land\beta_p\ge 1/3$; \par if $0\le p\le 6/5$ then: $\alpha_pA^p+(1- \alpha_p)G^p<I^p< \beta_pA^p+(1- \beta_p)G^p \Leftrightarrow \alpha_p\le 2/3 \land\beta_p\ge (2/e)^p$; \par if $p\ge 2$ then: $\alpha_pA^p+(1- \alpha_p)G^p<I^p< \beta_pA^p+(1- \beta_p)G^p \Leftrightarrow \alpha_p\le (2/e)^p \land\beta_p\ge 2/3$. \par $A,G, L, I$ 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.
[Peter S. Bullen (Vancouver)]

MSC 2000:: *26E60 Means
26D07 Inequalities involving other types of real functions

Keywords: geometric mean; logarithmic mean; identric mean; arithmetic mean; best constants

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