Cauchy estimates

Cauchy estimates bound the derivatives of a holomorphic function using its maximum magnitude on a circle contained in its domain.

Cauchy's integral formula: A formula expressing a holomorphic function's interior values as a contour integral over a surrounding curve. Differentiating this formula yields the derivative bounds.
Liouville's theorem: A theorem stating that every bounded entire function is constant. Applying estimates on circles of arbitrarily large radius forces every positive-order derivative to vanish.
Open disk: The set of complex points whose distance from a fixed center is less than a fixed radius. A disk gives the standard geometry for choosing the contour and measuring available radius.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point agree throughout a connected domain. Derivative control supports the local power-series structure behind this uniqueness result.
Maximum modulus principle: A theorem stating that a nonconstant holomorphic function cannot attain its maximum modulus inside a domain. It helps replace circle-wide values with boundary maxima in applications of the estimates.
Taylor series: A power series whose coefficients encode a function's derivatives at a point. Derivative bounds control Taylor coefficients and help establish convergence within a disk.
Factorial: For a nonnegative integer, the product of all positive integers up to that integer, with zero factorial equal to one. The nth-derivative bound contains n factorial from repeated differentiation of the integral formula.
Cauchy inequalities: Coefficient bounds for a holomorphic function's Taylor series obtained from Cauchy's integral formula. These closely related bounds target Taylor coefficients rather than derivatives directly.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity supplies the integral formula and the derivative structure being bounded.
Schwarz lemma: A theorem bounding holomorphic maps from the unit disk to itself that fix the origin. Cauchy estimates provide derivative control that underlies closely related disk bounds.

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