关于四元数典范广义多项式的Turan型不等式
On Turan Type inequality for Quaternionic Canonical Generalized Polynomials
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中文总结 AI 辅助
本文将Govil的经典多项式不等式从复情形推广到四元数框架，证明了满足零点代数条件的四元数典范广义多项式的Turan型不等式，为这类多项式提供了精确导数估计，推动经典不等式向非交换四元数领域的拓展。
中文摘要 AI 辅助
本文针对所有零点都位于半径k≥1的闭单位球内的四元数典范广义多项式，建立了Turan型不等式。我们将Govil提出的经典不等式从复情形推广到四元数框架。证明了对于某些类别的多项式，不等式‖P′‖≥n/(1+kⁿ)‖P‖对所有k≥1成立，其中n是多项式的次数。我们的主要结果为满足零点特定代数条件的四元数典范广义多项式提供了精确的导数估计，这些发现有助于将经典多项式不等式推广到四元数这一非交换情形的 ongoing 研究中。
英文摘要
This paper establishes Turán-type inequalities for quaternionic canonical generalized polynomials with all zeros in the closed unit ball of radius $k \geq 1$. We extend the classical inequality due to Govil from the complex setting to the quaternionic framework. We prove that for certain classes of polynomials, the inequality \[\|P'\| \geq \frac{n}{1+k^n}\|P\|\] holds for all $k \geq 1$, where $n$ is the degree of the polynomial. Our main results provide sharp derivative estimates for quaternionic canonical generalized polynomials under specific algebraic conditions on their zeros. These findings contribute to the ongoing research of extending classical polynomial inequalities to the noncommutative setting of quaternions.
