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 .
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.
© 1996-9 Eric W. Weisstein
1999-05-26