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

The authors note that the inequality

a 1 p - j=2 n a j p b 1 p - j=2 n b j p a 1 b 1 - j=2 n a j b j p

does not always hold for p1 (in particular not always for p>2) under the assumptions a j >0,b j >0(j=1,,n),a 1 p - j=2 n a j p >0,b 1 p - 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

a 1 p - j=2 n a j p b 1 p - j=2 n b j p n 1-min{2/p,1} a 1 b 1 - j=2 n a j b j p

under the above assumptions, and generalizations.

26D15Inequalities for sums, series and integrals of real functions
11E10Forms over real fields