Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Cauchy's inequalities bound every derivative by a boundary bound on a compactly contained circle
Statement
Let be holomorphic on and let . Suppose and
Then, for every ,
Facts & Assumptions
Given: A function holomorphic on , a radius , a bound on the radius- circle, and a natural number .
The higher-derivative Cauchy formula gives on the positively oriented radius- circle (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle).
The ML estimate bounds the modulus of a contour integral by a bound for the integrand times the contour length (ML estimate: a contour integral is bounded by a supremum bound times path length).
The once-traversed circle of radius has length (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).
Proof
On the circle, , so [L1], [L2], and [L3] give .
Since , simplifying step 1.1 gives . For this is , and for or constant the same computation remains valid.
Depends on
Used by
- Cauchy's inequalities bound the Taylor coefficients by the circle supremum Corollary
- A bounded function of two real variables whose every coordinate slice is real analytic is continuous False statement
- A bounded separately holomorphic function on a polydisc is Lipschitz on every smaller polydisc Lemma
- Why local boundedness gives joint continuity here and nothing like it holds in the real case Remark
- An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise Theorem
- Cauchy estimates for mixed derivatives on a polydisc Theorem
- Liouville's theorem: every bounded entire function is constant Theorem
Dependency tree · two levels
19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Lars Ahlfors, Complex Analysis, third edition, Ch. 4, Section 2.3 (standard reference, not scraped)