## 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)

If 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
 (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 .

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.