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 $ℂ$. 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., $f^{\prime }=\frac{\partial f}{\partial z}$ and $f^{(n)}=(f^{(n-1)}{)}^{\text{'}}$ (see Template:Section link).

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 $r\to \mathrm{\infty }$ 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 $|z-a|<r^{\prime }$,

$f(z)=\frac{1}{2\pi i}\underset{|w-a|=r^{\prime }}{\int }\frac{f(w)}{w-z}dw$

where $r^{\prime }<r$. By the differentiation under the integral sign (in the complex variable), we get:

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

Thus,

$|f^{(n)}(a)|\le \frac{n!M}{2\pi }\underset{|w-a|=r^{\prime }}{\int }\frac{|dw|}{|w-a{|}^{n+1}}=\frac{n!M}{{r^{\prime }}^n}.$

Letting $r^{\prime }\to r$ finishes the proof. $◻$

(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 $U\subset ℂ$, a compact subset $K\subset U$ and an integer $n>0$, there is a constant $C$ such that for every holomorphic function $f$ on $U$,

$\underset{K}{\sup }|f^{(n)}|\le C\underset{U}{\int }|f|d\mu$

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 $\partial u/\partial \bar{z}=f\partial \psi /\partial \bar{z}$, by the integral formula,

$u(z)=\frac{1}{2\pi i}\underset{\partial U}{\int }\frac{u(z)}{w-z}dw+\frac{1}{2\pi i}\underset{U}{\int }\frac{f(w)\partial \psi /\partial \bar{w}(w)}{w-z}dw\wedge d\bar{w}$

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 $\partial \psi /\partial \bar{w}$ 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 $U\subset ℂ$ 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 $U=\prod _1^{n}B(a_j,r_j)\subset {ℂ}^n$, we have: for each multiindex $\alpha \in {\mathbb{N}}^n$,

$|\left({\frac{\partial }{\partial z}}^{\alpha }f\right)(a)|\le \frac{\alpha !}{r^{\alpha }}\underset{U}{\sup }|f|$

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. Template:Section link and its consequence also continue to be valid in several variables with the same proofs.

See also

• Taylor's theorem

References

• Template:Citation
• Template:Cite book

Further reading

• https://math.stackexchange.com/questions/114349/how-is-cauchys-estimate-derived/114363

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