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Generalizations of Aczél’s inequality and Popoviciu’s inequality. (English) Zbl 1083.26019

The authors note that the inequality

$\left({a}_{1}^{p}-\sum _{j=2}^{n}{a}_{j}^{p}\right)\left({b}_{1}^{p}-\sum _{j=2}^{n}{b}_{j}^{p}\right)\le {\left({a}_{1}{b}_{1}-\sum _{j=2}^{n}{a}_{j}{b}_{j}\right)}^{p}$

does not always hold for $p\ge 1$ (in particular not always for $p>2$) under the assumptions ${a}_{j}>0,\phantom{\rule{0.166667em}{0ex}}{b}_{j}>0\phantom{\rule{4pt}{0ex}}\left(j=1,\cdots ,n\right),\phantom{\rule{4pt}{0ex}}{a}_{1}^{p}-{\sum }_{j=2}^{n}{a}_{j}^{p}>0,\phantom{\rule{4pt}{0ex}}{b}_{1}^{p}-{\sum }_{j=2}^{n}{b}_{j}^{p}>0$ stated in “Analytic inequalities” (1970; Zbl 0199.38101), pp. 58–59, of D. S. Mitrinović. (Note: This error has been noticed also by M. Bjelica [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 19, 105–109 (1990; Zbl 0733.26011)], and by L. Losonczi and Zs. Páles [J. Math. Anal. Appl. 205, No. 1, 148–156 (1997; Zbl 0871.26012)]. The latter also generalized the corrected inequality.

The present authors offer among others the inequality

$\left({a}_{1}^{p}-\sum _{j=2}^{n}{a}_{j}^{p}\right)\left({b}_{1}^{p}-\sum _{j=2}^{n}{b}_{j}^{p}\right)\le {\left({n}^{1-min\left\{2/p,1\right\}}{a}_{1}{b}_{1}-\sum _{j=2}^{n}{a}_{j}{b}_{j}\right)}^{p}$

under the above assumptions, and generalizations.

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions 11E10 Forms over real fields
##### Keywords:
Aczél inequality; Popoviciu inequality