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, yV, |(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, zX, d(x, y) + d(y, z) ≥ d(x, z).