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
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
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
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