Cauchy's estimate

Let f be a holomorphic function on the open ball B(a, r) in \mathbb C. If M is the sup of |f| over B(a, r), then Cauchy's estimate says: for each integer n > 0, :|f^{(n)}(a)| \le \frac{n!}{r^n} M where f^{(n)} is the n-th complex derivative of f; i.e., f' = \frac{\partial f}{\partial z} and f^{(n)} = (f^{(n-1)})^' (see ). Moreover, taking f(z) = z^n, a = 0, r = 1 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 r \to \infty in the estimate.) Slightly more generally, if f is an entire function bounded by A + B|z|^k for some constants A, B and some integer k > 0, then f is a polynomial. == Proof ==

We start with Cauchy's integral formula applied to f, which gives for z with | z - a | , :f(z) = \frac{1}{2\pi i} \int_{|w - a|^{n+1}} = \frac{n!M}{{r'}^n}. Letting r' \to r finishes the proof. \square (The proof shows it is not necessary to take M 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 M.) == Related estimate ==

Here is a somehow more general but less precise estimate. It says: given an open subset U \subset \mathbb{C}, a compact subset K \subset U and an integer n > 0, there is a constant C such that for every holomorphic function f on U, :\sup_{K} |f^{(n)}| \le C \int_U |f| \, d\mu where d\mu is the Lebesgue measure. This estimate follows from Cauchy's integral formula (in the general form) applied to u =\psi f where \psi is a smooth function that is =1 on a neighborhood of K and whose support is contained in U. Indeed, shrinking U, assume U is bounded and the boundary of it is piecewise-smooth. Then, since \partial u / \partial \overline{z} = f \partial \psi / \partial \overline{z}, by the integral formula, :u(z) = \frac{1}{2\pi i} \int_{\partial U} \frac{u(z)}{w - z} \, dw + \frac{1}{2\pi i} \int_U \frac{f(w) \partial \psi/\partial \overline{w} (w)}{w - z} \, dw \wedge d\overline{w} for z in U (since K can be a point, we cannot assume z is in K). Here, the first term on the right is zero since the support of u lies in U. Also, the support of \partial \psi/\partial \overline{w} is contained in U - K. 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 U \subset \mathbb{C} 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. == In several variables ==

Cauchy's estimate is also valid for holomorphic functions in several variables. Namely, for a holomorphic function f on a polydisc U = \prod_{j=1}^n B(a_j, r_j) \subset \mathbb{C}^n, we have: for each multiindex \alpha \in \mathbb{N}^n, :\left |\left(\frac{\partial}{\partial z}^{\alpha} f\right) (a) \right| \le \frac{\alpha!}{r^{\alpha}} \sup_U |f| where a = (a_1, \dots, a_n), \alpha! = \prod {\alpha}_j! and r^{\alpha} = \prod r_j^{\alpha_j}. 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. == See also ==