Dubinin–Sugawa (2009): the dual mean value inequality

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SmaleMeanValue.dubinin_sugawa_dual

by Lucas · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

complex-analysispolynomials

Let $P$ be a complex polynomial of degree $n \ge 2$ and let $z \in \mathbb C$ with $P'(z) \neq 0$ . Then there is a critical point $c$ of $P$ with

\left| \frac{P(z) - P(c)}{z - c} \right| \ge \frac{|P'(z)|}{n\,4^n}.

This is the reverse of the mean value inequality; optimizing the constant $\frac{1}{n4^n}$ is the dual mean value problem.

Preamble

import Mathlib
open Polynomial

Formal statement

namespace SmaleMeanValue

theorem dubinin_sugawa_dual (P : ℂ[X]) (hP : 2 ≤ P.natDegree) (z : ℂ)
    (hz : P.derivative.eval z ≠ 0) :
    ∃ c : ℂ, P.derivative.eval c = 0 ∧
      ‖P.derivative.eval z‖ / ((P.natDegree : ℝ) * 4 ^ P.natDegree) ≤
        ‖(P.eval z - P.eval c) / (z - c)‖ := by sorry

end SmaleMeanValue

Source

Wikipedia, "Mean value problem", revision oldid=1374678764, https://en.wikipedia.org/w/index.php?title=Mean_value_problem&oldid=1374678764, section "Partial results" (Dubinin–Sugawa, reverse inequality); Dubinin, Sugawa, Proc. Japan Acad. Ser. A 85(9) (2009), 135–137, https://arxiv.org/abs/0906.4605

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; NON-BLIND

Disclosure — NOT an independent read-back. This read-back is non-blind: it was written by the same agent that drafted the Lean statement, with full knowledge of the source material and of the intended meaning. It must not be treated as independent testimony. Reviewers should compare the Lean code against the source themselves.

Let $P$ be any polynomial with complex coefficients, and write $d$ for its degree (the natural-number degree; for the zero polynomial this would be $0$ , but here it is excluded because $d \ge 2$ is assumed). Write $P'$ for the formal derivative of $P$ and $P(w)$ for evaluation at $w \in \mathbb C$ . Let $z$ be any complex number such that $P'(z) \neq 0$ . The statement asserts that there exists $c \in \mathbb C$ with $P'(c) = 0$ and

\frac{|P'(z)|}{d \cdot 4^{d}} \le \left| \frac{P(z) - P(c)}{z - c} \right| ,

where $d \cdot 4^d$ is computed in the real numbers and is positive since $d \ge 2$ . This is a lower bound on the quotient for some critical point (existential, not for all critical points). Here $c$ ranges over all complex numbers; the condition $P'(c) = 0$ says that $c$ is a critical point of $P$ . Because $P'(z) \neq 0$ and $P'(c) = 0$ , necessarily $c \neq z$ , so the denominator $z - c$ is never zero and no division-by-zero convention is involved. The absolute values are the usual modulus on $\mathbb C$ .

Human review

Endorsed by Shuze Chen · Sep 30, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Sep 30, 2026
Confirmed by the mission captain (proposal self-audit).

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