Sharp Schoenberg type inequalities and the de Bruin--Sharma problem

In this paper, we confirm two conjectures proposed by Georgiev, Gómez-Serrano, Tao, and Wagner~\cite{GGTW25} on Schoenberg type inequalities of order $4$, thereby providing a complete solution to the de Bruin–Sharma problem. We further develop a new interpolation framework to study Schoenberg type inequalities and, in particular, give a new proof of Pereira’s result. Motivated by Sendov’s conjecture, we then derive sharp Schoenberg type inequalities of orders $-1$ and $-2m$ (with $m\in \mathbb{N}$), as well as non-sharp inequalities valid for all negative orders $p\le -1$. Finally, we discuss a dual counterpart of the Schoenberg type inequalities of order $-2$.