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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.

MSC:
26D15 Inequalities for sums, series and integrals
26E60 Means
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[1] Bullen, P.S.; Mitrinović, D.S.; Vasić, P.M., Means and their inequalities, (1988), Reidel Dordrecht · Zbl 0687.26005
[2] Hardy, G.; Littlewood, J.E.; Pólya, G., Inequalities, (1952), Cambridge Univ. Press Cambridge, UK · Zbl 0634.26008
[3] Mitrinović, D.S.; Vasić, P.M., Analytic inequalities, (1970), 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 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 New York · Zbl 0437.26007
[11] S.H. Wu, Some results on extending and sharpening the Weierstrass product inequalities, J. Math. Anal. Appl. (2005), in press · Zbl 1068.26024
[12] Robert, A.W.; Varberg, D.E., Convex function, (1973), Academic Press New York
[13] Mitrinović, D.S.; Pečarić, J.E.; Volenec, V., Recent advances in geometric inequalities, (1989), 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 Groningen · Zbl 0174.52401
[15] Ali, M.M., On some extremal simplexes, Pacific J. math., 33, 1-14, (1970) · Zbl 0197.16902
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