A special case of Hölder's sum inequality with
(1)
where equality holds for
(2)
In two-dimensions, it becomes
(3)
It can be proven by writing
(4)
If real complex
and can be found using the quadratic equation
(5)
In order for this to be complex , it must be true
that
(6)
with equality when vector derivation is much simpler,
(7)
where
(8)
and similarly for
See also Chebyshev Inequality ,
Chebyshev Sum Inequality ,
Hölder's
Inequalities ,
Schwarz's Inequality
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References Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Apostol, T. M. Calculus,
2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Cauchy, A. L. Cours
d'analyse de l'École Royale Polytechnique, 1ère partie: Analyse algébrique. uvres complètes,
2e série, Vol. 3. Paris: Gauthier-Villars, 1897. Gradshteyn,
I. S. and Ryzhik, I. M. Tables
of Integrals, Series, and Products, 6th ed. Hardy, G. H.; Littlewood, J. E.; and Pólya,
G. "Cauchy's Inequality." §2.4 in Inequalities,
2nd ed. Jeffreys,
H. and Jeffreys, B. S. "Cauchy's Inequality." §1.16 in Methods
of Mathematical Physics, 3rd ed. Krantz, S. G. Handbook
of Complex Variables. Mitrinović,
D. S. "Cauchy's and Related Inequalities." §2.6 in Analytic
Inequalities. Referenced
on Wolfram|Alpha Cauchy's Inequality
Cite this as:
Weisstein, Eric W. "Cauchy's Inequality."
From MathWorld https://mathworld.wolfram.com/CauchysInequality.html
Subject classifications
,
. The inequality is sometimes also called Lagrange's
inequality (Mitrinović 1970, p. 42), and can be written in vector form
as
is a constant 
,
then 
.
If it is not a constant, then all terms cannot simultaneously vanish for real 
, so the solution is complex
and can be found using the quadratic equation
is a constant. The vector derivation is much simpler,
.
