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m IraeVid moved page Inequality of arithmetic and geometric means to AM-GM Inequality: Misspelled: This inequality is more well-known as AM-GM Inequality. |
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One stronger version of this, which also gives strengthened version of the unweighted version, is due to Aldaz. In particular,
There is a similar inequality for the [[weighted arithmetic mean]] and [[weighted geometric mean]]. Specifically, let the nonnegative numbers {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, . . . , ''x<sub>n</sub>''}} and the nonnegative weights {{math|''w''<sub>1</sub>, ''w''<sub>2</sub>, . . . , ''w<sub>n</sub>''}} be given. Assume further that
the sum of the weights is 1. Then <math>\sum_{i=1}^n w_ix_i \geq \prod_{i=1}^n x_i^{w_i} + \sum_{i=1}^n w_i\left(x_i^{\frac{1}{2}} -\sum_{i=1}^n w_ix_i^{\frac{1}{2}} \right)^2 </math>. <ref>{{cite journal |last1=Aldaz |first1=J.M. |title=Self-Improvement of the Inequality Between Arithmetic and Geometric Means |journal=Journal of Mathematical Inequalities |date=2009 |volume=3 |issue=2 |page=213-216 |doi=10.7153/jmi-03-21 |url=http://jmi.ele-math.com/03-21/Self-improvement-of-the-inequality-between-arithmetic-and-geometric-means |access-date=11 January 2023|doi-access=free }}</ref>
====Proof using Jensen's inequality====
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