Conjugate index

In mathematics, two real numbers $p,q>1$ are called conjugate indices (or Hölder conjugates) if

$\frac{1}{p}+\frac{1}{q}=1.$

Formally, we also define $q=\mathrm{\infty }$ as conjugate to $p=1$ and vice versa.

Conjugate indices are used in Hölder's inequality, as well as Young's inequality for products; the latter can be used to prove the former. If $p,q>1$ are conjugate indices, the spaces L^p and L^q are dual to each other (see L^p space).

Properties

The following are equivalent characterizations of Hölder conjugates:

• $\frac{1}{p}+\frac{1}{q}=1,$
• $pq=p+q,$
• $\frac{p}{q}=p-1,$
• $\frac{q}{p}=q-1.$

See also

• Beatty's theorem

References

• Antonevich, A. Linear Functional Equations, Birkhäuser, 1999. Template:ISBN.

This article incorporates material from Conjugate index on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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