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 $q=\infty$ 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. ISBN 3-7643-2931-9.

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