Iterated Bernstein-type $L_p$ Inequalities for Polynomials

Abstract

We develop several new Bernstein-type $L_p$ inequalities for complex polynomials by iterating the first-order differential operator $A_\alpha(P):=zP'(z)-\alpha P(z)$. Our results extend, unify, and sharpen $L_p$ inequalities of Zygmund, de Bruijn, and Jain as well as the recent $L_p$ extensions for $A_\alpha$ and its second-order companion shown in \cite{RatherBhatGulzar2024}. In particular, for any finite sequence $\alpha_1,\dots,\alpha_m$ with $\Rea(\alpha_j)\le n/2$ we obtain sharp bounds for $\norm{\prod_{j=1}^m A_{\alpha_j} P}_p$ in terms of $\norm{P}_p$ for all $0\le p\le\infty$, together with refined ``Erd\H{o}s--Lax''-type improvements when $P$ has no zeros in the open unit disc. As corollaries, we derive $L_p$-versions of higher order Bernstein inequalities for $z^k P^{(k)}$ and scale-invariant formulations on circles $\{|z|=r\}$.