Borel–Carathéodory theorem

The Borel–Carathéodory theorem bounds a holomorphic function’s modulus on a smaller disk using an upper bound for its real part on a larger disk and the function’s value at the center.

Maximum modulus principle: A holomorphic function on a bounded domain attains its maximum modulus on the boundary unless it is constant. The theorem’s proof applies this principle to a transformed function whose boundary values are controlled by the real-part bound.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity is the essential hypothesis that lets real-part information constrain the full function.
Analytic function growth: The study of how the size of holomorphic functions changes across a domain or along expanding regions. The theorem converts control of real-part growth into a bound on the function itself.
Émile Borel: A French mathematician whose work contributed to measure theory, probability, and analysis. His name is attached to the theorem’s standard form.
Schwarz lemma: A holomorphic map from the unit disk to itself that fixes zero satisfies a sharp bound on its modulus. After normalization, the estimate uses disk self-maps and a Schwarz-lemma-type bound.
Harmonic function: A twice-differentiable function whose Laplacian vanishes, with local mean-value behavior. The real part of every holomorphic function is harmonic, supplying the theorem’s scalar control.
Derivative bounds for holomorphic functions: Upper bounds on derivatives of holomorphic functions obtained from domain-wide size or growth information. Its modulus estimate can be combined with Cauchy estimates to bound derivatives.
Constantin Carathéodory: A Greek-German mathematician known for major contributions to analysis, calculus of variations, and thermodynamics. His name is attached to the theorem, which appears in classical complex-analysis texts.
Cauchy estimates: Bounds on the derivatives of a holomorphic function obtained from its values on a surrounding contour. Combining this theorem with Cauchy estimates bounds derivatives from real-part data.
Real part: The real coordinate of a complex number, obtained by projecting it onto the real axis. The theorem uses an upper bound on this component to bound the function’s modulus.

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