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Symmetry 2018, 10(9), 380; https://doi.org/10.3390/sym10090380

The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

1
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
2
School of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economic, Chengdu, Sichuan 611130, China
3
Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands
4
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China
*
Author to whom correspondence should be addressed.
Received: 1 August 2018 / Revised: 25 August 2018 / Accepted: 27 August 2018 / Published: 4 September 2018
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Abstract

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies. View Full-Text
Keywords: weighted AM–GM inequality; Hölder inequality; weighted power-mean inequality; L’Hospital’s rule weighted AM–GM inequality; Hölder inequality; weighted power-mean inequality; L’Hospital’s rule
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Li, Y.; Gu, X.-M.; Zhao, J. The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality. Symmetry 2018, 10, 380.

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