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Theorem: Inequality of Weighted Arithmetic Mean

Let $p_1,\ldots,p_n$ be positive real numbers ("weights") and $a_1,\ldots,a_n$ be given real numbers. Then the weighted arithmetic mean can be approximated using the following inequality:

$$\min(a_1,\ldots,a_n)\le \frac {p_1a_1+\cdots+p_na_n}{p_1+\cdots+p_n} \le\max(a_1,\ldots,a_n),$$ where $\min$ and $\max$ denote the minimum and the maximum of the numbers $a_1,\ldots,a_n.$

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References

Bibliography

1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition