Conjugate index

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

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

Formally, we also define ${\displaystyle q=\infty }$ as conjugate to ${\displaystyle p=1}$ 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 ${\displaystyle p,q>1}$ 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:

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

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.