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

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School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
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School of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economic, Chengdu, Sichuan 611130, China
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Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands
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College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China
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Author to whom correspondence should be addressed.
Symmetry 2018, 10(9), 380; https://doi.org/10.3390/sym10090380
Received: 1 August 2018 / Revised: 25 August 2018 / Accepted: 27 August 2018 / Published: 4 September 2018
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
MDPI and ACS Style

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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