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Zbl 1083.26019
Wu, Shanhe; Debnath, Lokenath
Generalizations of Aczél's inequality and Popoviciu's inequality. (English)
[J] Indian J. Pure Appl. Math. 36, No. 2, 49-62 (2005). ISSN 0019-5588; ISSN 0975-7465/e

The authors note that the inequality $$\biggl(a_1^p-\sum^n_{j=2} a_j^p\biggr) \biggl(b_1^p-\sum^n_{j=2} b_j^p\biggr)\leq \biggl(a_1 b_1-\sum^n_{j=2} a_j b_j\biggr)^p$$ does not always hold for $p\geq 1$ (in particular not always for $p>2$) under the assumptions $a_j>0,\, b_j>0\ (j=1,\dots,n),\ a_1^p-\sum^n_{j=2} a_j^p >0,\ b_1^p-\sum^n_{j=2} b_j^p>0$ stated in ``Analytic inequalities'' (1970; Zbl 0199.38101), pp. 58--59, of {\it D. S. Mitrinović}. (Note: This error has been noticed also by {\it M. Bjelica} [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 19, 105--109 (1990; Zbl 0733.26011)], and by {\it L. Losonczi} and {\it 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 $$\biggl(a_1^p-\sum^n_{j=2} a_j^p\biggr) \biggl(b_1^p-\sum^n_{j=2} b_j^p\biggr)\leq \biggl(n^{1-\min\{2/p,1\}}a_1 b_1-\sum^n_{j=2} a_j b_j\biggr)^p$$ under the above assumptions, and generalizations.
[János Aczél (Waterloo/Ontario)]

MSC 2000:: *26D15 Inequalities for sums, series and integrals of real functions
11E10 Forms over real fields

Keywords: Aczél inequality; Popoviciu inequality

Citations: Zbl 0199.38101; Zbl 0733.26011; Zbl 0871.26012

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