Cauchy's estimate
In mathematics, specifically in complex analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function. These bounds are optimal.
Cauchy's estimate is also called Cauchy's inequality, but must not be confused with
the Cauchy–Schwarz inequality.
Statement and consequence
Let be a holomorphic function on the open ball in. If is the sup of over, then Cauchy's estimate says: for each integer,where is the n-th complex derivative of ; i.e., and .
Moreover, taking 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 Slightly more generally, if is an entire function bounded by for some constants and some integer, then is a polynomial.
Proof
We start with Cauchy's integral formula applied to, which gives for with,where. By the differentiation under the integral sign, we get:
Thus,
Letting finishes the proof.
Related estimate
Here is a somehow more general but less precise estimate. It says: given an open subset, a compact subset and an integer, there is a constant such that for every holomorphic function on,where is the Lebesgue measure.
This estimate follows from Cauchy's integral formula applied to where is a smooth function that is on a neighborhood of and whose support is contained in. Indeed, shrinking, assume is bounded and the boundary of it is piecewise-smooth. Then, since, by the integral formula,
for in . Here, the first term on the right is zero since the support of lies in. Also, the support of is contained in. 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, 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. 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.
In several variables
Cauchy's estimate is also valid for holomorphic functions in several variables. Namely, for a holomorphic function on a polydisc, we have: for each multiindex,where, and.
As in the one variable case, this follows from Cauchy's integral formula in polydiscs. and its consequence also continue to be valid in several variables with the same proofs.