“Cauchy Inequality”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cauchy Inequality

释义

Cauchy Inequality

A special case of the Hölder Sum Inequality with ,

(1)

where equality holds for

. In 2-D, it becomes

(2)

It can be proven by writing

(3)

is a constant

, then

. If it is not a constant, then all terms cannot simultaneously vanish forReal

, so the solution is Complex and can be found using theQuadratic Equation

(4)

In order for this to be Complex, it must be true that

(5)

with equality when

is a constant. The Vector derivation is much simpler,

(6)

where

(7)

and similarly for

See also Chebyshev Inequality, Hölder Sum Inequality

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

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