A Double Inequality for the Trigamma Function and Its Applications
We prove that p=1 and q=2 are the best possible parameters in the interval (0,∞) such that the double inequality ep/x+1-e-p/x/2p<ψ′x+1<eq/x+1-e-q/x/2q holds for x>0. As applications, some new approximation algorithms for the circumference ratio π and Catalan constant G=∑n=0∞-1n/(2n+1)2 are...
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Hindawi Limited
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/702718 |
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doaj-e4bc424c79b24e6d8e963db4402424332020-11-25T02:46:55ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/702718702718A Double Inequality for the Trigamma Function and Its ApplicationsZhen-Hang Yang0Yu-Ming Chu1Xiao-Jing Tao2School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, ChinaSchool of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, ChinaCollege of Mathematics and Econometrics, Hunan University, Changsha 410082, ChinaWe prove that p=1 and q=2 are the best possible parameters in the interval (0,∞) such that the double inequality ep/x+1-e-p/x/2p<ψ′x+1<eq/x+1-e-q/x/2q holds for x>0. As applications, some new approximation algorithms for the circumference ratio π and Catalan constant G=∑n=0∞-1n/(2n+1)2 are given. Here, ψ′ is the trigamma function.http://dx.doi.org/10.1155/2014/702718 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Zhen-Hang Yang Yu-Ming Chu Xiao-Jing Tao |
| spellingShingle |
Zhen-Hang Yang Yu-Ming Chu Xiao-Jing Tao A Double Inequality for the Trigamma Function and Its Applications Abstract and Applied Analysis |
| author_facet |
Zhen-Hang Yang Yu-Ming Chu Xiao-Jing Tao |
| author_sort |
Zhen-Hang Yang |
| title |
A Double Inequality for the Trigamma Function and Its Applications |
| title_short |
A Double Inequality for the Trigamma Function and Its Applications |
| title_full |
A Double Inequality for the Trigamma Function and Its Applications |
| title_fullStr |
A Double Inequality for the Trigamma Function and Its Applications |
| title_full_unstemmed |
A Double Inequality for the Trigamma Function and Its Applications |
| title_sort |
double inequality for the trigamma function and its applications |
| publisher |
Hindawi Limited |
| series |
Abstract and Applied Analysis |
| issn |
1085-3375 1687-0409 |
| publishDate |
2014-01-01 |
| description |
We prove that p=1 and q=2 are the best possible
parameters in the interval (0,∞) such that the double inequality ep/x+1-e-p/x/2p<ψ′x+1<eq/x+1-e-q/x/2q holds for x>0. As applications, some new approximation algorithms for the circumference ratio π and Catalan constant G=∑n=0∞-1n/(2n+1)2 are given. Here, ψ′ is the trigamma function. |
| url |
http://dx.doi.org/10.1155/2014/702718 |
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