Schmeisser’s conjecture (weighted barycenter to critical point)

Prove that for every n ≥ 2, the optimal constant C_schmeisser(n) equals 1; equivalently, for any degree-n complex polynomial with zeros z_1,…,z_n in the unit disk and nonnegative weights l_1,…,l_n summing to 1, there exists a critical point w_j of f′ with distance at most 1 from the weighted barycenter ∑_{k=1}^n l_k z_k.

Background

Schmeisser’s conjecture strengthens Sendov’s conjecture by controlling critical points relative to arbitrary convex combinations of zeros. While special cases and related results exist, the full conjecture remains open.

Resolution would deepen understanding of how zero configurations govern the geometry of critical points in complex polynomials.