mean_value_problem_leq_4

The following weaker version of the mean value problem has been proven. Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ 4|p'(z)|$.

lemma mean_value_problem_leq_4 (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‖ ≤ 4 *‖p.derivative.eval z‖

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