In mathematics, two real numbers p , q > 1 {\displaystyle p,q>1} are called conjugate indices (or Hölder conjugates) if
1 p + 1 q = 1. {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.}Formally, we also define q = ∞ {\displaystyle q=\infty } as conjugate to p = 1 {\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 p , q > 1 {\displaystyle p,q>1} are conjugate indices, the spaces Lp and Lq are dual to each other (see Lp space).
Properties
The following are equivalent characterizations of Hölder conjugates:
- 1 p + 1 q = 1 , {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1,}
- p q = p + q , {\displaystyle pq=p+q,}
- p q = p − 1 , {\displaystyle {\frac {p}{q}}=p-1,}
- q p = q − 1. {\displaystyle {\frac {q}{p}}=q-1.}
See also
- 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.