Cauchy's Inequality

Cauchy’s Inequality


an inequality for finite sums having the form

It is one of the most important and commonly used inequalities and was proved by A. Cauchy (1821). The integral analog of Cauchy's inequality was established by the Russian mathematician V. la. Buniakovskii, while an important generalization of Cauchy's inequality was made by the German mathematician L. O. Hölder.

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c) For p = q = 2 in inequality (10), we obtain a refinement of Cauchy's inequality
and after using Cauchy's inequality, the estimate 1/k+n < 1/n, and the definition of the moments in (1.
With Lemma 1 and Cauchy's inequality, we get (for t [greater than or equal to] 2)