Generalization and sharpness of the power means inequality and their applications
Copyright policy )Generalization and sharpness of the power means inequality and their applications
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 \textit{W. Janous, M. K. Kuczma} and \textit{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.
inequalities, Applied Mathematics, majorization, Inequalities for sums, series and integrals, Schur-convexity, power means, Analysis, Means
inequalities, Applied Mathematics, majorization, Inequalities for sums, series and integrals, Schur-convexity, power means, Analysis, Means
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