In this paper, we prove some inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. We further obtain the polar derivative generalizations of the obtained results. The obtained results produce various inequalities that are sharper than the previous ones while taking into account the placement (absolute value) of the zeros and the extremal coefficients of the polynomial. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.

中文翻译：

---

关于复域中多项式的不等式

在本文中，我们证明了一些与单变量复系数多项式及其导数的超范数相关的不等式，当其零点有限制时。我们进一步得到所得结果的极性导数推广。考虑到零点的位置（绝对值）和多项式的极值系数，所获得的结果会产生比之前更尖锐的各种不等式。此外，还提供了一些具体的数值例子，表明在某些情况下，我们的结果获得的界限可能比以前已知的界限更清晰。