Abstract

Let \(W(\zeta)=(a_{0}+a_{1}\zeta+...+a_{n}\zeta^{n})\) be a polynomial of degree \(n\) having all its zeros in \(\mathbb{T}_{k}\cup\mathbb{E}^{-}_{k}\), \(k\geq 1\), then for every real or complex number \(\alpha\) with \(|\alpha|\geq 1+k+k^{n}\), Govil and McTume [7] showed that the following inequality holds

$$\max\limits_{\zeta\in\mathbb{T}_{1}}|D_{\alpha}W(\zeta)|\geq n\left(\frac{|\alpha|-k}{1+k^{n}}\right)||W||+n\left(\frac{|\alpha|-(1+k+k^{n})}{1+k^{n}}\right)\min\limits_{\zeta\in\mathbb{T}_{k}}|W(\zeta)|.$$

In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered.

中文翻译：

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复多项式保图兰型不等式算子的推广

摘要

设\(W(\zeta)=(a_{0}+a_{1}\zeta+...+a_{n}\zeta^{n})\)为次数为\(n\)的多项式，其中所有它的零位于\(\mathbb{T}_{k}\cup\mathbb{E}^{-}_{k}\) 、 \( k \geq 1\)，然后对于每个实数或复数\( \alpha\)和\(|\alpha|\geq 1+k+k^{n}\)，Govil 和 McTume [7] 表明以下不等式成立

$$\max\limits_{\zeta\in\mathbb{T}_{1}}|D_{\alpha}W(\zeta)|\geq n\left(\frac{|\alpha|-k}{ 1+k^{n}}\right)||W||+n\left(\frac{|\alpha|-(1+k+k^{n})}{1+k^{n}} \right)\min\limits_{\zeta\in\mathbb{T}_{k}}|W(\zeta)|.$$

在本文中，我们获得了这种不等式的推广，涉及称为极导数的算子序列。此外，还考虑了极限情况的问题。