Abstract

As a generalization of a result obtained by Dubinin [7], Wali (preprint online) [14] recently proved the following: Let r ∈ \({{\mathcal{R}}_{n}}\), where r has n poles at a₁, a₂, …, a_n and all its zeros lie in |z| ≤ 1, with s-fold zeros at the origin, then for |z| = 1

$$\left| {r{\kern 1pt} '(z)} \right| \geqslant \frac{1}{2}\left\{ {\left| {\mathcal{B}{\kern 1pt} '(z)} \right| + (s + m - n) + \frac{{\left| {{{c}_{m}}} \right| - \left| {{{c}_{s}}} \right|}}{{\left| {{{c}_{m}}} \right| + \left| {{{c}_{s}}} \right|}}} \right\}\left| {r(z)} \right|.$$

In this paper, instead of assuming that r(z) has a zero of order s at the origin as Wali did, we suppose that r(z) has a zero of multiplicity s at any point inside the unit circle and all other zeros are inside or outside a circle of radius k. Further, we prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.

中文翻译：

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具有规定极点和限制零点的有理函数导数的不等式

摘要

作为 Dubinin [7] 所获得结果的推广，Wali（在线预印本）[14] 最近证明了以下内容： 令 r ∈ \({{\mathcal{R}}_{n}}\)，其中 r 有 a ₁、a ₂、...、a _n处的 n 个极点 及其所有零点位于| z | ≤ 1，在原点处有s折零，则对于| z | = 1

$$\左| {r{\kern 1pt} '(z)} \right| \geqslant \frac{1}{2}\left\{ {\left| {\mathcal{B}{\kern 1pt} '(z)} \right| + (s + m - n) + \frac{{\left| {{{c}_{m}}} \right| - \左| {{{c}_{s}}} \right|}}{{\left| {{{c}_{m}}} \right| + \左| {{{c}_{s}}} \right|}}} \right\}\left| {r(z)} \right|.$$

在本文中，我们没有像 Wali 那样假设r ( z ) 在原点处具有s阶零，而是假设r ( z )在单位圆内的任何点处具有重数s的零，并且所有其他零都是半径为k的圆的内部或外部。此外，我们证明了一些结果，除了概括有理函数的一些不等式之外，还包括对一些多项式不等式作为特殊情况的细化。