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

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

where f ( n ) {\displaystyle f^{(n)}} is the n-th complex derivative of f {\displaystyle f} ; i.e., f ′ = ∂ f ∂ z {\displaystyle f'={\frac {\partial f}{\partial z}}} and f ( n ) = ( f ( n − 1 ) ) ′ {\displaystyle f^{(n)}=(f^{(n-1)})^{'}} (see Wirtinger derivatives § Relation with complex differentiation). Moreover, taking f ( z ) = z n , a = 0 , r = 1 {\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 r → ∞ {\displaystyle r\to \infty } in the estimate.) Slightly more generally, if f {\displaystyle f} is an entire function bounded by A + B | z | k {\displaystyle A+B|z|^{k}} for some constants A , B {\displaystyle A,B} and some integer k > 0 {\displaystyle k>0} , then f {\displaystyle f} is a polynomial.

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

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

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

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

Thus,

| f ( n ) ( a ) | ≤ n ! M 2 π ∫ | w − a | = r ′ | d w | | w − a | n + 1 = n ! M r ′ n . {\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 r ′ → r {\displaystyle r'\to r} finishes the proof. ◻ {\displaystyle \square }

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

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

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

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

u ( z ) = 1 2 π i ∫ ∂ U u ( z ) w − z d w + 1 2 π i ∫ U f ( w ) ∂ ψ / ∂ w ¯ ( w ) w − z d w ∧ d w ¯ {\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 z {\displaystyle z} in U {\displaystyle U} (since K {\displaystyle K} can be a point, we cannot assume z {\displaystyle z} is in K {\displaystyle K} ). Here, the first term on the right is zero since the support of u {\displaystyle u} lies in U {\displaystyle U} . Also, the support of ∂ ψ / ∂ w ¯ {\displaystyle \partial \psi /\partial {\overline {w}}} is contained in U − K {\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, which says that a sequence of holomorphic functions on an open subset U ⊂ C {\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 f {\displaystyle f} on a polydisc U = ∏ j = 1 n B ( a j , r j ) ⊂ C n {\displaystyle U=\prod _{j=1}^{n}B(a_{j},r_{j})\subset \mathbb {C} ^{n}} , we have: for each multiindex α ∈ N n {\displaystyle \alpha \in \mathbb {N} ^{n}} ,

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

where ∂ ∂ z α = ∏ 1 n ( ∂ ∂ z j ) α j {\displaystyle {\frac {\partial }{\partial z}}^{\alpha }=\prod _{1}^{n}\left({\frac {\partial }{\partial z_{j}}}\right)^{\alpha _{j}}} , α ! = ∏ α j ! {\displaystyle \alpha !=\prod {\alpha }_{j}!} and r α = ∏ r j α j {\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.

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