Bloch's theorem (complex analysis)

Bloch's theorem states that every nonconstant holomorphic function on the unit disk maps some subdomain conformally onto a disk with radius at least a universal positive constant.

Bloch constant: The largest universal lower bound for the radius guaranteed by Bloch's theorem. It quantifies the theorem's guarantee and remains unknown exactly.
André Bloch: A French mathematician whose work includes the theorem on disks in images of holomorphic functions. His 1920s result introduced the universal image-disk phenomenon.
Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Bloch's assertion applies to this entire class of functions on the unit disk.
Bloch's constant problem: The problem of determining the exact optimal constant in Bloch's theorem. The theorem guarantees a positive radius, but its best possible value is not known.
Schwarz lemma: A result bounding holomorphic self-maps of the unit disk that fix the origin. Its normalization and derivative bounds are basic tools in related disk estimates.
Ahlfors' proof of Bloch's theorem: A proof of Bloch's theorem developed by Lars Ahlfors using covering-surface methods. It gives a notable approach to establishing the universal radius bound.
Unit disk: The set of complex numbers whose absolute value is less than one. It is the fixed source domain in the theorem's universal guarantee.
Ahlfors' conjecture: A conjecture predicting the exact value of Bloch's constant. It is the leading proposed resolution of Bloch's constant problem.
Koebe quarter theorem: Every normalized univalent function on the unit disk contains a disk of radius one quarter in its image. It gives a sharper image-disk guarantee when injectivity holds throughout the domain.
Landau's constant: The largest universal radius guaranteed by Landau's theorem for normalized holomorphic functions. Its extremal problem refines Bloch's image-radius question.

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