Power-mean M_{−2} bound in Sendov’s setting without additional assumptions

Determine whether, for every polynomial p(z) = (z − a) ∏_{j=1}^{n−1} (z − z_j) of degree n ≥ 2 with 0 < a ≤ 1 and |z_j| ≤ 1, the power mean of order −2 of the distances from the critical points to a, M_{−2}(|w_1 − a|, …, |w_{n−1} − a|) := ((1/(n−1)) Σ_{k=1}^{n−1} |w_k − a|^{−2})^{−1/2}, is at most 1 without any additional assumptions; an affirmative result would imply a conjecture stronger than Sendov’s original conjecture.

Background

Section 4 reformulates Sendov’s conjecture via power means of the distances from critical points to a given root a. The authors show that, under an extra assumption, a bound M_2 < 1 can be proved, but without that assumption M_2 < 1 fails in general.

They then observe that it remains unclear whether the stronger inequality M_{−2} ≤ 1 might always hold. Establishing M_{−2} ≤ 1 would be stronger than Sendov’s conjecture and could provide a new route to it, hence its significance.