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A simple proof on strengthening and extension of inequalities. (English) Zbl 1173.26329

The main result of the paper is that among the arithmetic mean \(A,\) the power exponential mean \(Z,\) and the contraharmonic mean \(C,\) we have the inequalities: \(A\leq Z\leq C\) thus, if \(a,b>0,\) then \[ \frac{a+b}{2}\leq a^{\frac{a}{a+b}}b^{\frac{b}{a+b}}\leq \frac{a^{2}+b^{2}}{ a+b}. \] Similar inequalities for means in \(n\) variables are also proved.

MSC:

26E60 Means
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
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