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 $f$ 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:^[1] for each integer $n>0$ ,

|f^{(n)}(a)|\leq {\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 Wirtinger derivatives § Relation with complex differentiation).

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.^[2]

Proof

We start with Cauchy's integral formula applied to $f$ , which gives for $z$ with $|z-a|<r'$ ,

f(z)={\frac {1}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{w-z}}\,dw

where $r'<r$ . By the differentiation under the integral sign (in the complex variable),^[3] we get:

f^{(n)}(z)={\frac {n!}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{(w-z)^{n+1}}}\,dw.

Thus,

|f^{(n)}(a)|\leq {\frac {n!M}{2\pi }}\int _{|w-a|=r'}{\frac {|dw|}{|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:^[4] 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)}|\leq 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,^[5] 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 $U=\prod _{1}^{n}B(a_{j},r_{j})\subset \mathbb {C} ^{n}$ , we have:^[6] for each multiindex $\alpha \in \mathbb {N} ^{n}$ ,

\left|\left({\frac {\partial }{\partial z}}^{\alpha }f\right)(a)\right|\leq {\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. § Related estimate and its consequence also continue to be valid in several variables with the same proofs.^[7]

References

^ Rudin 1986, Theorem 10.26.
^ Rudin 1986, Ch 10. Exercise 4.
^ This step is Exercise 7 in Ch. 10. of Rudin 1986
^ Hörmander 1990, Theorem 1.2.4.
^ Hörmander 1990, Corollary 1.2.6.
^ Hörmander 1990, Theorem 2.2.7.
^ Hörmander 1990, Theorem 2.2.3., Corollary 2.2.5.

Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables (3rd ed.), North Holland, ISBN 978-1-493-30273-4
Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill. ISBN 978-0-07-054234-1.