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Zbl 1083.26018
Wu, Shanhe
Generalization and sharpness of the power means inequality and their applications.
(English)
[J] J. Math. Anal. Appl. 312, No. 2, 637-652 (2005). ISSN 0022-247X

The main results of the paper sharpen the classical well-known inequalities between power means. As a consequence, the inequality $$\left(\sum_{i=1}^n x_i\right)^n \le (n-1)^{n-1} \sum_{i=1}^n x_i^n + n\big(n^{n-1}-(n-1)^{n-1}\big)\prod_{i=1}^n x_i$$ is proved for all $x_1,\dots,x_n>0$, $n\ge2$, which was conjectured by {\it W. Janous, M. K. Kuczma} and {\it 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.
[Zsolt Páles (Debrecen)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
26E60 Means

Keywords: power means; majorization; Schur-convexity; inequalities

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