Phragmén–Lindelöf principle

The Phragmén–Lindelöf principle is a family of theorems extending maximum principles to unbounded domains. Growth bounds prevent holomorphic or subharmonic functions from exceeding boundary constraints.

Subharmonic function: A function satisfying a mean-value inequality that generalizes the behavior of the logarithm of a holomorphic function's modulus. Subharmonic formulations extend the principle beyond holomorphic functions.
Phragmén–Lindelöf theorem for a sector: A sector theorem bounding a holomorphic function from its boundary values under a growth restriction tied to the sector angle. It is the canonical setting where angular width determines the critical growth rate.
Lars Edvard Phragmén: A Swedish mathematician known for work in complex analysis and for co-formulating the Phragmén–Lindelöf principle. His work with Lindelöf established the sector-based extension of maximum principles.
Paley–Wiener theorem: A family of theorems relating the support of a function to growth properties of its Fourier transform. Phragmén–Lindelöf estimates help establish growth bounds central to Fourier-transform results.
Schwarz lemma: A bound on holomorphic self-maps of the unit disk that fix the origin. It gives sharp control on a bounded disk, unlike the growth-dependent control needed here.
Harmonic measure: A probability measure describing how boundary values influence a harmonic function at an interior point. It encodes how boundary control competes with growth toward infinity.
Phragmén–Lindelöf theorem for a strip: A theorem controlling holomorphic functions in an infinite strip from bounds on its two boundary lines and suitable growth conditions. The strip illustrates boundary control across an unbounded domain with parallel edges.
Ernst Leonard Lindelöf: A Finnish mathematician whose work in complex analysis includes the Phragmén–Lindelöf principle. His collaboration with Phragmén gave the principle its name and early form.
Fourier transform: An integral transform expressing a function in terms of its frequency components. Complex growth estimates can constrain Fourier transforms and their inverses.
Liouville's theorem: Every bounded entire function is constant. It shows how a global boundedness condition alone controls functions on the whole plane.

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Carlson's theoremRelated: Related half-plane growth arguments propagate boundary bounds into the interior.