Knowlpedia

Theorem

Cauchy estimate for holomorphic derivatives

A bound on a complex neighborhood controls all derivatives on a smaller set with factorial constants.

Statement

If ff is holomorphic on a neighborhood of the closed disc D‾(a,R)\overline D(a,R), then

∣f(m)(a)∣≤m!Rmmax⁡∣z−a∣=R∣f(z)∣,m≥0.|f^{(m)}(a)|\le \frac{m!}{R^m}\max_{|z-a|=R}|f(z)|, \qquad m\ge0.

This Cauchy estimate follows from the derivative form of the : bound the integrand and use the circle length 2πR2\pi R.

Uniform neighborhoods

If every point of a compact real interval has its closed radius-RR disc in a common complex domain where ∣f∣≤M|f|\le M, then sup⁡I∣f(m)∣≤Mm!R−m\sup_I|f^{(m)}|\le M m!R^{-m} for every mm. The same domain supplies all orders at once. In a polydisc, iterating the formula gives ∣∂αf(a)∣≤α!M∏jRj−αj|\partial^\alpha f(a)|\le\alpha!M\prod_jR_j^{-\alpha_j}.

Room for polynomial factors

If a norm includes an extra factor such as (m+1)2(m+1)^2, choosing a strictly smaller radius absorbs it: for 0<r<R0<r<R, the sequence (m+1)2(r/R)m(m+1)^2(r/R)^m is bounded. A positive gap between radii is therefore useful even when each fixed derivative already has a Cauchy bound.

References