Math 2.0

Functions of a complex variable Geometric function theory Conjecture

Brennan conjecture

Builds on Riemann mapping theoremSimply connected domainConformal map

Statement

Let Ω⊊ℂ\Omega \subsetneq \C be a simply connected domain, and let φ ⁣:Ω→𝔻\varphi \colon \Omega \to \mathbb{D} be a conformal map onto the unit disc 𝔻={z∈ℂ:∣z∣<1}\mathbb{D} = \set{z \in \C : \abs{z} < 1}. Then for every real pp with 4/3<p<44/3 < p < 4, ∫Ω∣φ′(z)∣p dA(z)<∞,\int_\Omega \abs{\varphi'(z)}^p \, dA(z) < \infty, where dAdA is area measure.

Remarks

Such a map φ\varphi exists by the Riemann mapping theorem. For p=2p = 2 the integral is the area of 𝔻\mathbb{D}, which is π\pi. The range is sharp at both ends. Take the Koebe domain Ω=ℂ∖(−∞,−1/4]\Omega = \C \setminus (-\infty, -1/4] and the inverse φ\varphi of the Koebe function. Then ∣φ′(z)∣\abs{\varphi'(z)} behaves like ∣z∣−3/2\abs{z}^{-3/2} at infinity and like ∣z+1/4∣−1/2\abs{z + 1/4}^{-1/2} near the tip of the slit. So the integral diverges for p≤4/3p \le 4/3 and for p≥4p \ge 4. The conjecture is known for 4/3<p<3+δ4/3 < p < 3 + \delta with some δ>0\delta > 0; Brennan proved this range. The full conjecture is open.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptConformal map, concept, no evidence levelConformal mapconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSimply connected domain, concept, no evidence levelSimply connected doma…conceptReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionConformal map, definition, no evidence levelConformal mapdefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionRiemann mapping theorem, statement, no evidence levelRiemann mapping theor…statementBrennan conjecture, conjecture, no evidence levelBrennan conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Complex plane concept
  3. Conformal map concept
  4. Real numbers object
  5. Set bounded above definition
  6. Simply connected domain concept
  7. Real and imaginary parts definition
  8. Conformal map definition
  9. Supremum definition
  10. Completeness axiom statement
  11. Existence of square roots of nonnegative reals statement
  12. Square root definition
  13. Modulus of a complex number definition
  14. Complex differentiability definition
  15. Holomorphic function definition
  16. Riemann mapping theorem statement
  17. Brennan conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Brennan_conjecture
About this page
TagCH73TVRT27
URIm2://CH73TVRT27
Kindconjecture
WritingW0 stub
MSC30C35, 30C55
StatusThis copy of the site has no build data. Run m2 status CH73TVRT27.
AboutConformal map
UsesRiemann mapping theorem, Simply connected domain, Conformal map
Same as
Changed2 November 2026

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