A special case of the Hölder Sum Inequality with ,

(1) 
where equality holds for . In 2D, 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 .
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.
© 19969 Eric W. Weisstein
19990526