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.
Highlights
In the field of classical analysis, the weighted arithmetic mean–geometric mean (AM–GM)inequality is often inferred from Jensen’s inequality, which is a more generalized inequality compared to the AM–GM inequality; refer to, e.g., [1,2]
These studies note the relations among the weighted AM–GM inequality, the Hölder inequality, and the weighted power-mean inequality are still less clear, one inequality is often helpful to prove another inequality [1,12]
The Hölder inequality is equivalent to the weighted AM–GM inequality
Summary
In the field of classical analysis, the weighted arithmetic mean–geometric mean (AM–GM)inequality (see e.g., [1], pp. 74–75) is often inferred from Jensen’s inequality, which is a more generalized inequality compared to the AM–GM inequality; refer to, e.g., [1,2]. The Hölder inequality [2] These studies note the relations among the weighted AM–GM inequality, the Hölder inequality, and the weighted power-mean inequality are still less clear, one inequality is often helpful to prove another inequality [1,12]. Motivated by these aforementioned studies, in the present note, the mathematical equivalence among three such well-known inequalities is proved in detail; the result introduced in [14] is extended.
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