The Hadamard Three-Lines Theorem is a result of complex analysis describing the behavior of a bounded holomorphic function defined on a strip bounded by two parallel vertical lines in the complex plane. It states that the maximum modulus of such a function on any intermediate vertical line within the strip is controlled by, and interpolates logarithmically between, its maximum moduli on the two boundary lines. Named for the French mathematician Jacques Hadamard, the theorem is a foundational interpolation result used to prove later convexity theorems in analysis, including the Riesz-Thorin interpolation theorem for operators on Lp spaces.
Facts
StatementFor a bounded function holomorphic in the interior of a strip and continuous on the whole strip, with M(x) the supremum of its modulus on the vertical line at x, log M(x) is a convex function on [a,b]. 1 Classification
Statement Form Connections
Has Statement Form
In Branch
Source Hadamard three-lines theorem, Wikipedia
Proved By
Source Hadamard three-lines theorem, Wikipedia
Sources
1. Hadamard three-lines theorem, Wikipedia
Statement
then log M(x) is a convex function on [a,b].
In Branch: Complex Analysis, Lead sentence
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic funct
Proved By: Jacques Hadamard, Lead section
The theorem is named after the French mathematician Jacques Hadamard.
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.