I was playing around with Smale's Mean Value Conjecture and found a curious formulation of it which would be stronger (and which may simply be false). It seems to hold for `generic' random polynomials but of course this may mean very little. I'd be interested in whether there is a chance it might be true or whether there are counterexamples (it isn't really clear to me how to try to construct one).

**Smale's Mean Value Conjecture.** Smale proved in 1981 that if $f:\mathbb{C} \rightarrow \mathbb{C}$ is a polynomial with $f(0) = 0$ and $|f'(0)| = 1$, then there exists a critical point of the polynomial ($z \in \mathbb{C}$ such that $f'(z) = 0$) satisfying
$$ |f(z)| \leq 4 |z|.$$
One question is whether 4 can be replaced by 1 (which would be best possible if one is interested in universal constants that do not depend on the degree). The best bound depending on the degree is conjectured to be $(d-1)/d$.

I was looking at some examples and noticed a curious pattern which seems to hold at least for polynomials with randomly generated roots with very high probability (meaning I did not find a counterexample but maybe random polynomials are the wrong place to look).

**A Stronger Conjecture?** Let $g:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial with $|g(0)|=1$. Consider the subset
$$ A = \left\{z \in \mathbb{C}: |g(z)| < 1 \right\} \subset \mathbb{C}$$
and let $B$ denote the connected component of $A$ whose closure contains the point 0. Is it true that the number of critical points of $z g(z)$ in $B$ is exactly the same as the number of roots of $z g(z)$ in $B$? Or, a weaker formulation, that $B$ contains at least one critical point?

Note that $\log |g(z)|$ is analytic and constant on the boundary, therefore $B$ is simply connected and $B$ contains at least one root of $g(z)$. If this statement were true, it would therefore imply Smale's conjecture. It would also be a bit finer since Smale's conjecture is a priori a statement about the set $A$ containing a critical point (and $B \subset A$).

The subsequent picture gives an example: the red dots are the roots of $z g(z)$, the black dots represent critical points of $z g(z)$. The set $A$ is shown in blue, the set $B$ is the big blue set in the middle. There is one root of $z g(z)$ on the boundary of $B$ (that is the origin). The set then contains 4 roots and 4 critical points.

This seems to be true at least for polynomials with randomly chosen roots. A more complicated example is shown below. What is nice about the formulation is that it seems somewhat more tractable: the function $g$ maps the set $B$ to the unit disk and the boundary to the boundary. So the question is really whether $g(z)$ and $g(z) + z g'(z)$ have the same number of roots. I like this formulation because it is much closer to the usual Rouche-type questions and seems a little bit easier to approach (if it were true).

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