Mean value problem for polynomial critical points

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $P\in\mathbb C[z]$ be a polynomial of degree $d\ge2$, and let $z\in\mathbb C$ satisfy $P'(z)\ne0$. A critical point is a complex number $c$ such that $P'(c)=0$. For such $z$, no critical point equals $z$, so the quantity \[\mu(P,z)=\min_{P'(c)=0}\left|\frac{P(c)-P(z)}{(c-z)P'(z)}\right|\] is well defined. Repeated critical points do not affect this minimum. Smale's known mean-value estimate gives $\mu(P,z)\le4$; the usual sharper conjecture asks for $\mu(P,z)\le1-1/d$.

Statement recorded in the supplied source.

Does the bound $\mu(P,z)\le1$ hold for every polynomial $P$ of degree at least two and every point $z$ at which $P'(z)\ne0$? Equivalently, must there exist a critical point $c$ such that \[|P(c)-P(z)|\le |P'(z)|\,|c-z|?\] [S2]

Short English statement

For any complex polynomial and any noncritical point, is there a critical point whose secant slope in absolute value is at most the derivative at the given point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2400007 (AMR-023-0007). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Edward Crane, Mean Value Conjectures for Rational Maps (2004), arXiv:math/0411604, Section 1, p. 1, Smale theorem and equation (1). https://arxiv.org/pdf/math/0411604.