关于Ankeny和Rivlin定理锐化的注解

IF 1 4区数学 Q1 MATHEMATICS

Applicable Analysis and Discrete Mathematics Pub Date : 2021-01-01 DOI:10.2298/AADM200206012D

Aseem Dalal, K. Govil

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引用次数: 0

摘要

令p(z) = ?n?= 0 z ?是n次多项式，M(p,R) = max|z|=R?0 |p(z)|， and M(p,1):= ||p||。然后根据Ankeny和Rivlin的一个著名的结果，我们有R ?1 M(p,R) ?p (Rn + 1/2) | | | |。这种不平等被Govil强化了，他证明了R ?1 M(p,R) ?p (Rn + 1/2) | | | | - n / 2 (| | p | | 2 - 4 | | 2 / | | p | |) {(r1) | | p | |和| | p | | + 2 | | - ln (1 + (r1) | | p | |和| | p | | + 2 | |)}。在本文中，我们锐化了Govil的上述不等式，这反过来锐化了Ankeny和Rivlin的不等式。我们用LerchPhi函数?(z,s,a)来表示我们的结果，该函数在Wolfram的MATHEMATICA中实现为LerchPhi [z,s,a]，它可以计算为任意数值精度，并且适用于符号和数值操作。此外，我们还给出了一个例子，并通过MATLAB证明了对某些多项式的界的改进是相当显著的。

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A note on sharpening of a theorem of Ankeny and Rivlin

Let p(z) = ?n?=0 a?z? be a polynomial of degree n, M(p,R) := max|z|=R?0 |p(z)|, and M(p,1) := ||p||. Then according to a well-known result of Ankeny and Rivlin, we have for R ? 1, M(p,R) ? (Rn+1/2) ||p||. This inequality has been sharpened among others by Govil, who proved that for R ? 1, M(p,R) ? (Rn+1/2) ||p||-n/2 (||p||2-4|an|2/||p||) {(R-1)||p||/||p||+2|an|- ln (1+ (R-1)||p||/||p||+2|an|)}. In this paper, we sharpen the above inequality of Govil, which in turn sharpens inequality of Ankeny and Rivlin. We present our result in terms of the LerchPhi function ?(z,s,a), implemented in Wolfram's MATHEMATICA as LerchPhi [z,s,a], which can be evaluated to arbitrary numerical precision, and is suitable for both symbolic and numerical manipulations. Also, we present an example and by using MATLAB show that for some polynomials the improvement in bound can be considerably significant.

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来源期刊

Applicable Analysis and Discrete Mathematics MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.40

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： Applicable Analysis and Discrete Mathematics is indexed, abstracted and cover-to cover reviewed in: Web of Science, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), Mathematical Reviews/MathSciNet, Zentralblatt für Mathematik, Referativny Zhurnal-VINITI. It is included Citation Index-Expanded (SCIE), ISI Alerting Service and in Digital Mathematical Registry of American Mathematical Society (http://www.ams.org/dmr/).