# Cauchy inequality

## Cauchy inequality

[kō·shē ‚in·i′kwäl·əd·ē]
(mathematics)
The square of the sum of the products of two variables for a range of values is less than or equal to the product of the sums of the squares of these two variables for the same range of values.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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By virtue of [(18).sub.1], (53), Lemmas A.1-A.4, Cauchy inequality, and Holder inequality, we get
From (53), (56), Lemmas A.1-A.4, Cauchy inequality, and Hoolder inequality, we get
b) For p = q = 2, we deduce a refinement for the integral version of the Cauchy inequality can be formulated as follows:
Further, by (2.4), (2.5), the Cauchy inequality, (3.3) and Lemma 3.4, we get
Finally, (2.4), the Cauchy inequality and Lemma 3.4 imply that
Hence, by Cauchy inequality, [parallel][??](t)[parallel]]' [less than or equal to] [parallel][??](t)[parallel] for all t [member of] J := {t [member of] [0,T] : [parallel][??](t)[parallel] > 0}.
From the Cauchy inequality and the Prime Theorem (see references , ,  and ) we may get

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