Abstract

This paper considers the well known Erdős–Lax and Turán-type inequalities that relate the uniform norm of a univariate complex coefficient polynomial to that of its derivative on the unit circle in the plane. Here, we establish some new inequalities that relate the uniform norm of a polynomial and its polar derivative while taking into account the placement of the zeros and the extremal coefficients of the polynomial. The obtained results strengthen some recently proved Erdős–Lax and Turán-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in the literature on this subject.

中文翻译：

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多项式及其极导数的极值问题

摘要

本文考虑了众所周知的 Erdős-Lax 和 Turán 型不等式，它们将单变量复系数多项式的一致范数与其在平面单位圆上的导数的一致范数联系起来。在这里，我们建立了一些新的不等式，这些不等式将多项式的一致范数及其极导数联系起来，同时考虑多项式的零点位置和极值系数。所获得的结果强化了一些最近证明的约束多项式的 Erdős-Lax 和 Turán 型不等式，并且还产生了比该主题文献中已知的先前不等式更尖锐的各种不等式。