Generalization and sharpness of the power means inequality and their applications. (English) Zbl 1083.26018

The main results of the paper sharpen the classical well-known inequalities between power means. As a consequence, the inequality \[ \left(\sum_{i=1}^n x_i\right)^n \leq (n-1)^{n-1} \sum_{i=1}^n x_i^n + n\big(n^{n-1}-(n-1)^{n-1}\big)\prod_{i=1}^n x_i \] is proved for all \(x_1,\dots,x_n>0\), \(n\geq2\), which was conjectured by W. Janous, M. K. Kuczma and M. S. Klamkin [Problem 1598, Crux Math. 16, 299–300 (1990), per bibl.]. The methods of the paper are analytic and use majorization and Schur-convexity. Some geometric applications are also obtained.


26D15 Inequalities for sums, series and integrals
26E60 Means
Full Text: DOI


[1] Bullen, P. S.; Mitrinović, D. S.; Vasić, P. M., Means and Their Inequalities (1988), Reidel: Reidel Dordrecht · Zbl 0687.26005
[2] Hardy, G.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0634.26008
[3] Mitrinović, D. S.; Vasić, P. M., Analytic Inequalities (1970), Springer-Verlag: Springer-Verlag New York · Zbl 0319.26010
[4] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0771.26009
[5] Zhang, W. P.; Yi, Y., On a sum analogous to the Dedekind sum and its first power mean value formula, J. Math. Anal. Appl., 256, 542-555 (2001) · Zbl 0972.11089
[6] Liu, Z., Note on generalization of power means and their inequalities, J. Math. Anal. Appl., 237, 726-729 (1999) · Zbl 0938.26009
[7] Pečarić, J. E., Generalization of the power means and their inequalities, J. Math. Anal. Appl., 161, 395-404 (1991) · Zbl 0753.26009
[8] Farnsworth, D.; Orr, R., Transformation of power means and a new class of means, J. Math. Anal. Appl., 129, 394-400 (1988) · Zbl 0638.26015
[9] Janous, W.; Kuczma, M. K.; Klamkin, M. S., Problem 1598, Crux Math., 16, 299-300 (1990)
[10] Marshall, A. W.; Olkin, I., Inequalities: The Theory of Majorization and Its Applications (1979), Academic Press: Academic Press New York · Zbl 0437.26007
[12] Robert, A. W.; Varberg, D. E., Convex Function (1973), Academic Press: Academic Press New York
[13] Mitrinović, D. S.; Pečarić, J. E.; Volenec, V., Recent Advances in Geometric Inequalities (1989), Kluwer Academic: Kluwer Academic Dordrecht, pp. 463-473 · Zbl 0679.51004
[14] Bottema, O.; Djordjević, R. Z.; Janić, R. R.; Mitrinović, D. S.; Vasić, P. M., Geometric Inequalities (1969), Wolters-Noordhoff: Wolters-Noordhoff Groningen · Zbl 0174.52401
[15] Ali, M. M., On some extremal simplexes, Pacific J. Math., 33, 1-14 (1970) · Zbl 0197.16902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.