In mathematics, specifically in complex analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function.
Cauchy's estimate is also called Cauchy's inequality, but must not be confused with the Cauchy–Schwarz inequality.
be a holomorphic function on the open ball
, then Cauchy's estimate says:[1] for each integer
is the n-th complex derivative of
(see Wirtinger derivatives § Relation with complex differentiation).
shows the above estimate cannot be improved.
As a corollary, for example, we obtain Liouville's theorem, which says a bounded entire function is constant (indeed, let
is an entire function bounded by
[2] We start with Cauchy's integral formula applied to
By the differentiation under the integral sign (in the complex variable),[3] we get: Thus, Letting
finishes the proof.
(The proof shows it is not necessary to take
to be the sup over the whole open disk, but because of the maximal principle, restricting the sup to the near boundary would not change
Here is a somehow more general but less precise estimate.
is the Lebesgue measure.
This estimate follows from Cauchy's integral formula (in the general form) applied to
is bounded and the boundary of it is piecewise-smooth.
, by the integral formula, for
can be a point, we cannot assume
Here, the first term on the right is zero since the support of
Thus, after the differentiation under the integral sign, the claimed estimate follows.
As an application of the above estimate, we can obtain the Stieltjes–Vitali theorem,[5] which says that a sequence of holomorphic functions on an open subset
that is bounded on each compact subset has a subsequence converging on each compact subset (necessarily to a holomorphic function since the limit satisfies the Cauchy–Riemann equations).
Indeed, the estimate implies such a sequence is equicontinuous on each compact subset; thus, Ascoli's theorem and the diagonal argument give a claimed subsequence.
Cauchy's estimate is also valid for holomorphic functions in several variables.
As in the one variable case, this follows from Cauchy's integral formula in polydiscs.
§ Related estimate and its consequence also continue to be valid in several variables with the same proofs.
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