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Cauchy estimates on a smaller concentric disc

Statement

Let $0 \leq r < R < S$ , let $f$ be holomorphic on $D (a, S)$ , and suppose $M \geq 0$ satisfies $∣ f (ζ) ∣ \leq M$ whenever $∣ ζ - a ∣ = R$ . If $n \in N$ and $∣ z - a ∣ \leq r$ , then

$∣ f^{(n)} (z) ∣ \leq \frac{n! R M}{(R - r)^{n + 1}} .$

Facts & Assumptions

Given: Reals $0 \leq r < R < S$ , a function $f$ holomorphic on $D (a, S)$ , a bound $M \geq 0$ on the radius- $R$ circle, a natural $n$ , and a point $z$ with $∣ z - a ∣ \leq r$ .

[L1]

Cauchy's higher-derivative formula on the radius- $R$ circle gives $f^{(n)} (z) = n! (2 π i)^{- 1} \int_{γ} f (ζ) / (ζ - z)^{n + 1} d ζ$ for $∣ z - a ∣ < R < S$ (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle).

[L2]

Complex modulus is multiplicative and satisfies the triangle inequality (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L3]

If an integrand has modulus at most $B$ on a rectifiable contour $γ$ , then the modulus of its integral is at most $B L (γ)$ (ML estimate: a contour integral is bounded by a supremum bound times path length).

[L4]

A once-traversed circle of radius $R > 0$ has length $2 π R$ (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).

Proof

technique · direct

1.1L1L2

Formula [L1] applies because $∣ z - a ∣ \leq r < R$ , and for $∣ ζ - a ∣ = R$ the triangle inequality in [L2] gives $∣ ζ - z ∣ \geq ∣ ζ - a ∣ - ∣ z - a ∣ \geq R - r > 0$ , so the integrand has modulus at most $M / (R - r)^{n + 1}$ .

2.1step 1.1L3L4algebra

Applying [L3] to step 1.1 and using the circle length from [L4] gives $∣ f^{(n)} (z) ∣ \leq (n! / (2 π)) (M / (R - r)^{n + 1}) (2 π R) = n! R M / (R - r)^{n + 1}$ .

3.1step 2.1∎

The bound in step 2.1 is independent of $z$ on the closed radius- $r$ disc and includes derivative order $n = 0$ , inner radius $r = 0$ , and bound $M = 0$ ; the strict inequality $r < R$ keeps every denominator positive.

Cauchy estimates on a smaller concentric disc

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources