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 ${\displaystyle f}$ be a holomorphic function on the open ball ${\displaystyle B(a,r)}$ in ${\displaystyle \mathbb {C} }$. If ${\displaystyle M}$ is the sup of ${\displaystyle |f|}$ over ${\displaystyle B(a,r)}$, then Cauchy's estimate says: ^[1] for each integer ${\displaystyle n>0}$,

${\displaystyle |f^{(n)}(a)|\leq {\frac {n!}{r^{n}}}M}$

where ${\displaystyle f^{(n)}}$ is the n-th complex derivative of ${\displaystyle f}$; i.e., ${\displaystyle f'={\frac {\partial f}{\partial z}}}$ and ${\displaystyle f^{(n)}=(f^{(n-1)})^{'}}$ (see Wirtinger derivatives § Relation with complex differentiation).

Moreover, taking ${\displaystyle f(z)=z^{n},a=0,r=1}$ 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 ${\displaystyle r\to \infty }$ in the estimate.) Slightly more generally, if ${\displaystyle f}$ is an entire function bounded by ${\displaystyle A+B|z|^{k}}$ for some constants ${\displaystyle A,B}$ and some integer ${\displaystyle k>0}$, then ${\displaystyle f}$ is a polynomial. ^[2]

Proof

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

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

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

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

Thus,

${\displaystyle |f^{(n)}(a)|\leq {\frac {n!M}{2\pi }}\int _{|w-a|=r'}{\frac {|dw|}{|w-a|^{n+1}}}={\frac {n!M}{{r'}^{n}}}.}$

Letting ${\displaystyle r'\to r}$ finishes the proof. ${\displaystyle \square }$

(The proof shows it is not necessary to take ${\displaystyle M}$ 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 ${\displaystyle M}$.)

Related estimate

Here is a somehow more general but less precise estimate. It says: ^[4] given an open subset ${\displaystyle U\subset \mathbb {C} }$, a compact subset ${\displaystyle K\subset U}$ and an integer ${\displaystyle n>0}$, there is a constant ${\displaystyle C}$ such that for every holomorphic function ${\displaystyle f}$ on ${\displaystyle U}$,

${\displaystyle \sup _{K}|f^{(n)}|\leq C\int _{U}|f|\,d\mu }$

where ${\displaystyle d\mu }$ is the Lebesgue measure.

This estimate follows from Cauchy's integral formula (in the general form) applied to ${\displaystyle u=\psi f}$ where ${\displaystyle \psi }$ is a smooth function that is ${\displaystyle =1}$ on a neighborhood of ${\displaystyle K}$ and whose support is contained in ${\displaystyle U}$. Indeed, shrinking ${\displaystyle U}$, assume ${\displaystyle U}$ is bounded and the boundary of it is piecewise-smooth. Then, since ${\displaystyle \partial u/\partial {\overline {z}}=f\partial \psi /\partial {\overline {z}}}$, by the integral formula,

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

for ${\displaystyle z}$ in ${\displaystyle U}$ (since ${\displaystyle K}$ can be a point, we cannot assume ${\displaystyle z}$ is in ${\displaystyle K}$). Here, the first term on the right is zero since the support of ${\displaystyle u}$ lies in ${\displaystyle U}$. Also, the support of ${\displaystyle \partial \psi /\partial {\overline {w}}}$ is contained in ${\displaystyle U-K}$. 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 that a sequence of holomorphic functions on an open subset ${\displaystyle U\subset \mathbb {C} }$ 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 ${\displaystyle f}$ on a polydisc ${\displaystyle U=\prod _{1}^{n}B(a_{j},r_{j})\subset \mathbb {C} ^{n}}$, we have: ^[6] for each multiindex ${\displaystyle \alpha \in \mathbb {N} ^{n}}$,

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

where ${\displaystyle a=(a_{1},\dots ,a_{n})}$, ${\displaystyle \alpha$ !=\prod {\alpha }_{j}!} and ${\displaystyle r^{\alpha }=\prod r_{j}^{\alpha _{j}}}$.

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]

See also

• Taylor's theorem

References

1. Rudin 1986 , Theorem 10.26.
2. Rudin 1986 , Ch 10. Exercise 4.
3. This step is Exercise 7 in Ch. 10. of Rudin 1986
4. Hörmander 1990 , Theorem 1.2.4.
5. Hörmander 1990 , Corollary 1.2.6.
6. Hörmander 1990 , Theorem 2.2.7.
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.

Further reading

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