The monotonicity of ratios involving arc tangent function with applications
In this paper, we investigate the monotonicity of the functions on (0, ∞) for > 0, which not only gives relative errors of known bounds with quadratic for arctan , but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan is presente...
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| Published in | Open mathematics (Warsaw, Poland) Vol. 17; no. 1; pp. 1450 - 1467 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Warsaw
De Gruyter
01.01.2019
De Gruyter Poland |
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| ISSN | 2391-5455 2391-5455 |
| DOI | 10.1515/math-2019-0098 |
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| Abstract | In this paper, we investigate the monotonicity of the functions
on (0, ∞) for
> 0, which not only gives relative errors of known bounds with quadratic for arctan
, but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan
is presented. |
|---|---|
| AbstractList | In this paper, we investigate the monotonicity of the functions
on (0, ∞) for
> 0, which not only gives relative errors of known bounds with quadratic for arctan
, but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan
is presented. In this paper, we investigate the monotonicity of the functions In this paper, we investigate the monotonicity of the functions $$\begin{array}{} \begin{split}{} \displaystyle x &\mapsto &\frac{1}{x}\left( 1-a+\sqrt{\frac{2}{3}ax^{2}+a^{2}}\right) \arctan x, \\ x &\mapsto &\frac{1}{x}\left( \frac{4}{\pi ^{2}}+\sqrt{\frac{4}{\pi ^{2}}% x^{2}+a}\right) \arctan x \end{split} \end{array}$$ on (0, ∞) for a > 0, which not only gives relative errors of known bounds with quadratic for arctan x , but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan x is presented. In this paper, we investigate the monotonicity of the functionsx↦1x1−a+23ax2+a2arctanx,x↦1x4π2+4π2x2+aarctanxon (0, ∞) for a > 0, which not only gives relative errors of known bounds with quadratic for arctan x, but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan x is presented. |
| Author | Gao, Qin Yang, Zhen-Hang Tin, King-Fung |
| Author_xml | – sequence: 1 givenname: Zhen-Hang surname: Yang fullname: Yang, Zhen-Hang email: yzhkm@163.com organization: Zhejiang Society for Electric Power, 310014, Hangzhou, P. R. China – sequence: 2 givenname: King-Fung surname: Tin fullname: Tin, King-Fung email: tinkf_hbu@126.com organization: College of Science and Technology, North China Electric Power University, Ruixiang Street 282, 071051, Baoding, P. R. China – sequence: 3 givenname: Qin surname: Gao fullname: Gao, Qin email: nknkmdk@gmail.com organization: College of Science and Technology, North China Electric Power University, Ruixiang Street 282, 071051, Baoding, P. R. China |
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| CitedBy_id | crossref_primary_10_3390_axioms11060262 crossref_primary_10_1007_s13398_021_01152_x crossref_primary_10_3934_math_2020466 |
| Cites_doi | 10.1186/s13660-018-1704-0 10.1090/proc/14199 10.1155/2009/930294 10.4064/cm136-2-8 10.1016/j.jmaa.2015.03.043 10.1186/s13660-016-1157-2 10.1186/s13662-018-1545-7 10.1186/s13660-017-1312-4 10.1080/00029890.1966.11970755 10.1007/s13398-018-0609-6 10.1155/2011/840206 10.1016/j.jmaa.2014.05.034 10.2298/AADM1801244N 10.1186/s13660-018-1734-7 10.2307/2314285 |
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| Snippet | In this paper, we investigate the monotonicity of the functions
on (0, ∞) for
> 0, which not only gives relative errors of known bounds with quadratic for... In this paper, we investigate the monotonicity of the functions $$\begin{array}{} \begin{split}{} \displaystyle x &\mapsto &\frac{1}{x}\left(... In this paper, we investigate the monotonicity of the functionsx↦1x1−a+23ax2+a2arctanx,x↦1x4π2+4π2x2+aarctanxon (0, ∞) for a > 0, which not only gives... In this paper, we investigate the monotonicity of the functions |
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| SubjectTerms | absolute error arctangent function Primary 33B10 relative error Secondary 26D05 sharp bounds |
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| Title | The monotonicity of ratios involving arc tangent function with applications |
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