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单词 WeierstrassProductInequality
释义

Weierstrass product inequality


For any finite family (ai)iI of real numbers in the interval[0,1], we have

i(1-ai)1-iai.

Proof: Write

f=i(1-ai)+iai.

For any kI, and any fixed values of the ai for ik,f is a polynomialPlanetmathPlanetmath of the first degree in ak.Consequently f is minimal either at ak=0 or ak=1.That brings us down to two cases: all the ai are zero, or at leastone of them is 1. But in both cases it is clear that f1, QED.

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更新时间:2025/5/5 2:57:39