Mathematical
Inequalities| Mathematical
Inequalities |
Bernoulli Inequality
$ Def: If x 
–1,
then (1+x)n 1
+ nx
for all n 
N;
Can be proved by induction, or using the binomial theorem.
Cauchy-Schwarz Inequality
$ Def: The Schwarz inequality
in L2[a, b], i.e., for
any two functions f, g L2[a, b],
|
ab dx f *(x)
g(x)|2 
[ab dx |f(x)|2]
[ab dx |g(x)|2]
.
Minkowski Inequality
$ Def: For functions f, g Lp,
p > 1, 
f + g p f p +
g p.
* Relationships: It is
a consequence of the Hölder inequality,
and a generalization of the triangle inequality.
Schwarz Inequality
$ Def: If V is
an inner product space, 
x,
y V,
|(x, y)|2 (x, x)
(y, y).
@ References: Bhatia & Davis CMP(00) [operator versions].
Triangle Inequality
* In C: For all z1 and z2, z1
+ z2 z1 + z2 .
* In a general metric space:
For all x, y, z X, d(x, y)
+ d(y, z) d(x, z).
Other > see Hölder Inequality; Orlicz Inequality; quantum states.
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