This paper considers the well known Bernstein and Turán-type inequalities that relate the uniform norm of a 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 the polar derivative and the polynomial, in case when the zeros are inside or outside some closed disk. The obtained results produce various inequalities that are sharper than the previous ones known in very rich literature on this subject.

中文翻译：

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具有零约束的多项式的Bernstein和Turán型不等式

本文考虑了众所周知的伯恩斯坦和图兰型不等式，它们将多项式的统一范数与其在平面上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，它们将极导数和多项式的统一范数联系在一起，以防零点位于某个封闭圆盘的内部或外部。所获得的结果产生了各种不平等，这些不平等比在非常丰富的文献中有关该主题的先前不平等更为尖锐。