Hölder's inequality

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In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality relating L^p spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in L^p(S) and g be in L^q(S). Then fg is in L¹(S) and

$\|fg\|_1 \le \|f\|_p \|g\|_q.$

The numbers p and q above are said to be Hölder conjugates of each other.

Hölder's inequality is used to prove the generalization of the triangle inequality in the space L^p, the Minkowski inequality, and also to establish that L^p is dual to L^q.

Hölder's inequality was first found by L. J. Rogers (1888), and rediscovered by Hölder (1889).

1 Notable special cases
2 Generalization
3 Proof
4 References

[edit] Notable special cases

For p = q = 2 Holder's inequality results in the Cauchy-Schwarz inequality.

In the case of the Euclidean space, when set S is {1,...,n} with the counting measure, one has that for all x, y in Rⁿ (Cⁿ)

$\sum_{k=1}^n |x_k y_k| \leq \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} \left( \sum_{k=1}^n |y_k|^q \right)^{1/q}.$

If S = N with the counting measure, then we get Holder's inequality for sequences from lp spaces

$\sum\limits_{n=1}^{\infty} |x_n \cdot y_n| \le \left( \sum\limits_{n=1}^{\infty} |x_n|^p \right)^{1/p} \cdot \left( \sum\limits_{n=1}^{\infty} |y_n|^q \right)^{1/q},\; \forall x \in \ell^p, y\in \ell^q.$

For the space of integrable complex-valued functions, one has

$\left|\int f(x)g(x)\,dx\right|\leq \int \bigg| f(x)g(x)\bigg| \, dx \leq\left(\int \left|f(x)\right|^p\,dx \right)^{1/p}\cdot \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.$

For the probability space $(\Omega,\mathcal{F},\mathbb{P})$ , $L^p(\Omega,\mathcal{F},\mathbb{P})$ denotes the space of the random variables with finite p-moment, $\mathbb{E}\left[|X|^p\right] < \infty$ , where the symbol $\mathbb{E}$ denotes the expected value. Holder's inequality becomes

$\mathbb{E}|XY| \le \left(\mathbb{E}|X|^p\right)^{1/p} \cdot \left( \mathbb{E}|Y|^q \right)^{1/q},\; \forall X \in L^p, Y \in L^q.$

The inequality becomes equality when

| Y | = k | X | p - 1

for some constant k.

[edit] Generalization

The following generalization can be proven by induction.

Assume $p_k\geq 1, k=1,\ldots, n$ are such that

$\sum_{k=1}^n \frac{1}{p_k}=1.$

Assume that $u_k\in L^{p_k}(S)$ . Then $\prod_{k=1}^n u_k \in L^1(S)$ and

$\left\|\prod_{k=1}^n u_k\right\|_{\displaystyle L^1(S)}\leq \prod_{k=1}^n \|u_k\|_{\displaystyle L^{p_k}(S)}.$

[edit] Proof

We want to prove that

$\sum a_i b_i \le (\sum a_i^p)^{\frac {1}{p}} \cdot (\sum b_i^q)^{\frac {1}{q}}.$

We observe that both sides are homogeneous of degree 1 in a and b, so by multiplying the sequences a and b by constants we can assume that Σa^p = Σb^q = 1 (in which case the right hand side of the inequality is 1).

But this special case follows from Young's inequality ab ≤ a^p/p + b^q/q as follows:

$\sum a_ib_i \le \frac {1}{p} \sum a_i^p + \frac {1}{q} \sum b_i^q = \frac {1}{p} + \frac {1}{q} = 1.$

[edit] References

L.P. Kuptsov, "Hölder inequality" SpringerLink Encyclopaedia of Mathematics (2001)
O. Hölder, Ueber einen Mittelwerthsatz Nachr. Ges. Wiss. Göttingen (1889) pp. 38–47
G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities , Cambridge Univ. Press (1934)
L J. Rogers, An extension of a certain theorem in inequalities Messenger of math 17, 145-150.

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Categories: Inequalities | Functional analysis