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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 ${f}^{\text{'}}/{g}^{\text{'}}$ is increasing so are the ratios $\left(f\left(x\right)-f\left(b\right)\right)/\left(g\left(x\right)-g\left(b\right)\right),\left(f\left(x\right)-f\left(a\right)\right)/\left(g\left(x\right)-g\left(a\right)\right)$.

If $p\ge 1$ then: ${\alpha }_{p}{A}^{p}+\left(1-{\alpha }_{p}\right){G}^{p}<{L}^{p}<{\beta }_{p}{A}^{p}+\left(1-{\beta }_{p}\right){G}^{p}⇔{\alpha }_{p}\le 0\wedge {\beta }_{p}\ge 1/3$;

if $0\le p\le 6/5$ then: ${\alpha }_{p}{A}^{p}+\left(1-{\alpha }_{p}\right){G}^{p}<{I}^{p}<{\beta }_{p}{A}^{p}+\left(1-{\beta }_{p}\right){G}^{p}⇔{\alpha }_{p}\le 2/3\wedge {\beta }_{p}\ge {\left(2/e\right)}^{p}$;

if $p\ge 2$ then: ${\alpha }_{p}{A}^{p}+\left(1-{\alpha }_{p}\right){G}^{p}<{I}^{p}<{\beta }_{p}{A}^{p}+\left(1-{\beta }_{p}\right){G}^{p}⇔{\alpha }_{p}\le {\left(2/e\right)}^{p}\wedge {\beta }_{p}\ge 2/3$.

$A,G,L,I$ 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.

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