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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.

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