Convex-function extension of the order-1 Schoenberg-type inequality

Determine whether the order-1 Schoenberg-type inequality established in Theorem 3.4 under the centroid condition Σ_{j=1}^n z_j = 0, namely Σ_{k=1}^{n-1} |w_k| ≤ √(n−2) Σ_{j=1}^n |z_j| for zeros z_j and critical points w_k of a complex polynomial p(z) of degree n, remains valid when the absolute value is replaced by an arbitrary convex function φ: C → R; specifically, ascertain whether Σ_{k=1}^{n-1} φ(w_k) ≤ √(n−2) Σ_{j=1}^n φ(z_j) holds for all such polynomials.

Background

Theorem 3.4 introduces a Schoenberg-type inequality of order 1: under the centroid condition Σ z_j = 0, the sum of moduli of the critical points is bounded by a constant times the sum of moduli of the zeros. This strengthens a classical inequality of de Bruijn–Springer and Erdős–Niven, which Pereira later generalized by allowing an arbitrary convex function in place of the absolute value for inequality (8).

Remark 3.5 explicitly asks whether a similar convex-function generalization holds for the new order-1 inequality (10). This would extend the scope of Schoenberg-type inequalities beyond the absolute value, paralleling Pereira’s extension for the classical order-1 relation between zeros and critical points.