Mathematical Inequalities| Mathematical Inequalities |
In General > s.a. Inequalities [in physics];
for other mathematical ones, see Hölder Inequality;
Orlicz Inequality; quantum states.
@ References: Garling 07 [in linear analysis].
Bernoulli Inequality
> s.a. MathWorld page;
Wikipedia page.
$ Def: If x ≥ −1,
then (1+x)n ≥ 1 + nx
for all n ∈ \(\mathbb N\); It can be proved by induction, or using the
binomial theorem.
Cauchy or Cauchy-Schwarz Inequality
> s.a. quantum correlations.
$ Def: The Schwarz inequality
expressed for elements of 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] .
@ References:
Bhattacharyya a1907 [generalization].
> Online resources:
see MathWorld page;
Wikipedia page.
Minkowski Inequality
> s.a. MathWorld page;
Wikipedia page.
$ 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
> s.a. MathWorld page.
$ Def: If V is
an inner product space, then ∀x, y ∈ V,
|(x, y)|2 ≤ (x, x)
(y, y), or \(|{\bf x}\cdot{\bf y}| \le \Vert{\bf x}\Vert\,\Vert{\bf y}\Vert\).
@ References: Bhatia & Davis CMP(00) [operator versions].
Triangle Inequality > s.a. distance
and finsler geometry [reverse 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).
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