A note on sharpening of a theorem of Ankeny and Rivlin

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-01-01 DOI:10.2298/AADM200206012D

Aseem Dalal, K. Govil

引用次数: 0

Abstract

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.

查看原文本刊更多论文

关于Ankeny和Rivlin定理锐化的注解

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.