Source labels openChecked July 26, 2026

Mean value problem

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$ , such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$ .

Field theory and polynomials AMS 12

Mathematical statement

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$ , such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$ .

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

mean_value_problem

Canonical source

Complete statement target, proof intentionally absentLean 4

lemma mean_value_problem (p : Polynomial ℂ) (hp : 2 ≤ p.degree) (z : ℂ) (K : ℝ):    ∃ c : ℂ, p.derivative.eval c = 0 ∧      ‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ ‖p.derivative.eval z‖ := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wikipediabackground The fundamental theorem of algebra and complexity theorybackground Critical points and values of complex polynomialsbackground