If \(P(z)=a_n\prod \nolimits _{j=1}^n(z-z_j)\) is a complex polynomial of degree n having all its zeros in \(|z|\le K,K\ge 1\) then Aziz (Proc Am Math Soc 89:259–266, 1983) proved that$$\begin{aligned} \max _{|z|=1}{|P'(z)|}\ge \frac{2}{1+K^n}\sum _{j=1}^n\frac{K}{K+|z_j|}\max _{|z|=1}{|P(z)|}. \end{aligned}$$(0.1)In this paper we sharpen the inequality (0.1) and further extend the obtained result to the polar derivative of a polynomial. As a consequence we also derive two results on the generalization of Erdös–Lax type inequality for the class of polynomials having no zeros in the disc \(|z|<K,\;K\le 1\).

中文翻译：

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关于多项式的不等式

如果\（P（z）= a_n \ prod \ nolimits _ {j = 1} ^ n（z-z_j）\）是阶数为n的复数多项式，且所有零在\（| z | \ le K，K \ ge 1 \），然后Aziz（Proc Am Math Soc 89：259–266，1983）证明$$ \ begin {aligned} \ max _ {| z | = 1} {| P'（z）|} \ ge \ frac {2} {1 + K ^ n} \ sum _ {j = 1} ^ n \ frac {K} {K + | z_j |} \ max _ {| z | = 1} {| P（z）| }。\ end {aligned} $$（0.1）在本文中，我们扩大了不等式（0.1），并将所得结果进一步扩展到多项式的极导数。结果，对于圆盘\（| z | <K，\ ;; K \ le 1 \）中没有零的多项式，我们还得到了关于Erdös-Lax型不等式推广的两个结果。