Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros in \(|z|\ge k\) or in \(|z|\le k\), \(k>0\). In this paper, first we obtain inequalities about the dependence of |p(Rz)| on |p(rz)|, where \(|z|=1\), for \(r^2\le Rr\le k^2\) or \(k^2\le Rr \le R^2\). Further, another similar inequality for the class of polynomials having all zeros in \(|z|\ge k\), \(k>0\) is also proved for \(0<r\le R\le k\). Our results improve as well as generalize certain well-known polynomial inequalities.

中文翻译：

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有限零多项式的最大模数

设p ( z ) 是n 次多项式，其重数s在原点为零，其余零在\(|z|\ge k\)或\(|z|\le k\) , \(k >0\)。在本文中，首先我们获得关于 | 的依赖的不等式。p ( Rz )| 在 | p ( rz )|，其中\(|z|=1\)，对于\(r^2\le Rr\le k^2\)或\(k^2\le Rr \le R^2\)。此外，对于\(|z|\ge k\)中全为零的多项式类的另一个类似的不等式，\(k>0\)也证明了\(0<r\le R\le k\)。我们的结果改进并推广了某些众所周知的多项式不等式。