Local optimum for Sendov's conjecture

Question

For Sendov's conjecture, the distance 1 appears in the conjecture is tight, if one consider the polynomials $f_{n}(z) = z^{n} - 1$ for all $n\geq 2$. I wonder if this polynomial is the local optima for the conjecture. More precisely, I want to know if the following statement is true:

For all $n\geq 2$, there exists $\epsilon = \epsilon_{n} > 0$ such that for all degree $n$ polynomials $f(z) \in \mathbb{C}[z]$ with $\| f - f_{n}\| < \epsilon$, $f$ satisfies the Sendov's conjecture.

Here the norm is given as the $L^\infty$-norm of the coefficient vector ($\|a_{n}z^{n} + \cdots + a_{1}z + a_{0}\| = \max_{0\leq i\leq n} \{|a_{i}|\}$). I couldn't find any works on Sendov's conjecture in this direction.

Answer 1

This follows from the work of

Miller, Michael J., On Sendov’s conjecture for roots near the unit circle, J. Math. Anal. Appl. 175, No. 2, 632-639 (1993). ZBL0782.30007.

and independently

Vâjâitu, Viorel; Zaharescu, A., Ilyeff’s conjecture on a corona, Bull. Lond. Math. Soc. 25, No. 1, 49-54 (1993). ZBL0796.30004.

who established Sendov's conjecture when the distinguished zero is sufficiently close to the unit circle. By Rouche's theorem, any sufficiently small perturbation of $f_n$ will have its zeroes close enough to the unit circle for one of these two results to apply.

If one uses the more recent result of

Kasmalkar, Indraneel G., On the Sendov conjecture for a root close to the unit circle, Aust. J. Math. Anal. Appl. 11, No. 1, Article No. 4, 34 p. (2014). ZBL1293.30018.

then one can obtain an explicit value of $\epsilon_n$ for your question, probably of polynomial type in $n$ (although the asymptotic behavior in $n$ is not so relevant now, due to my recent result establishing the conjecture for all sufficiently large $n$).