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Inequalities for differences of powers. (English) Zbl 0649.26014

Generalizing a result by E. Leach and M. Sholander [same Journal 92, 207-223 (1983; Zbl 0517.26007)] the author presents necessary conditions for \[ \prod^{n}_{k=1}| x^{a_ k}-y^{a_ k}|^{q_ k}\quad \geq \prod^{n}_{k=1}| a_ k|^{q_ k}\quad (\sum^{n}_{k=1}q_ k=0) \] and shows that in the case \(k=4\), \(a_ 1\leq a_ 2\leq a_ 3\leq a_ 4\), \((a_ 1+a_ 4)(a_ 2+a_ 3)\geq 0\), they are also sufficient.
{Note: It seems that in (i) and (ii), on page 272, \(\leq\) should stand rather than =.}
Reviewer: J.Aczél

MSC:

26D15 Inequalities for sums, series and integrals
41A10 Approximation by polynomials
26A51 Convexity of real functions in one variable, generalizations
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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References:

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