Optimal Lehmer mean bounds for the $n$th power-type Toader means of n=-1,1,3

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-01-01 DOI:10.7153/jmi-2022-16-12

Tie-hong Zhao, Hong-Hu Chu, Yuming Chu

引用次数: 29

Abstract

In the article, we prove that λ1 = 0 , μ1 = 5/8 , λ2 = −1/8 , μ2 = 0 , λ3 = −1 and μ3 = −7/8 are the best possible parameters such that the double inequalities Lλ1 (a,b) < T3(a,b) < Lμ1 (a,b), Lλ2 (a,b) < T1(a,b) < Lμ2 (a,b), Lλ3 (a,b) < T−1(a,b) < Lμ3 (a,b) hold for a,b > 0 with a = b , and provide new bounds for the complete elliptic integral of the second kind E (r) = ∫ π/2 0 (1− r2 sin2 θ )1/2dθ on the interval (0,1) , where Lp(a,b) = (ap+1 + bp+1)/(ap +bp) is the p -th Lehmer mean and Tn(a,b) = ( 2 π ∫ π/2 0 √ an cos2 θ +bn sin2 θdθ )2/n is the n th power-type Toader mean. Mathematics subject classification (2020): 26E60, 33E05.

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第n次幂型Toader均值n=-1,1,3的最优Lehmer均值界

在这篇文章中,我们证明λ1 = 0,μ1 = 5/8,λ2 =−1/8,μ2 = 0,λ3 =−1和μ3 =−7/8是最好的参数,这样双重的不平等Lλ1 (a, b) < T3 (a, b) < Lμ1 (a、b), Lλ2 (a, b) < T1 (a, b) < Lμ2 (a, b), Lλ3 (a, b) < T−1 (a, b) < Lμ3 (a, b)保持b > 0 = b,并提供完整的新界限第二类椭圆积分E (r) =∫π/ 2 0(1−r2 sin2θ)dθ的间隔(0,1),其中Lp(a,b) = (ap+1 +bp +1)/(ap +bp)为p - Lehmer均值，Tn(a,b) = (2 π∫π/2 0√and cos2 θ +bn sin2 θdθ)2/n为n次幂型Toader均值。数学学科分类(2020):26E60, 33E05。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.