A generalization of the arithmeticgeometric means inequality
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 by A. M. Fink and Max Jodeit PDF
 Proc. Amer. Math. Soc. 61 (1976), 255261 Request permission
Abstract:
It is shown that the arithmetic mean of ${x_1}{w_1}, \ldots ,{x_n}{w_n}$ exceeds the geometric mean of ${x_1}, \ldots ,{x_n}$ unless all the $x$’s are equal, where ${w_1}, \ldots ,{w_n}$ depend on ${x_1}, \ldots ,{x_n}$ and satisfy $0 \leqslant {w_i} < 1$ unless ${x_i} = \min {x_k}$. This inequality is then applied to an integral inequality for functions $y$ defined on $[0,\;\infty )$ with ${y^{(k)}}$ convex and $0$ at $0$ for $0 \leqslant k < n$.References

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Additional Information
 © Copyright 1976 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 61 (1976), 255261
 MSC: Primary 26A86
 DOI: https://doi.org/10.1090/S00029939197604275642
 MathSciNet review: 0427564