Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality
Abstract Based on the Padé approximation method, in this paper we determine the coefficients a j $a_{j}$ and b j $b_{j}$ ( 1 ≤ j ≤ k $1\leq j \leq k$ ) such that 1 e ( 1 + 1 x ) x = x k + a 1 x k − 1 + ⋯ + a k x k + b 1 x k − 1 + ⋯ + b k + O ( 1 x 2 k + 1 ) , x → ∞ , $$ \frac{1}{e} \biggl( 1+\frac{1...
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doaj-65c03364ca3243e08f4a9716572b7d142020-11-25T00:54:43ZengSpringerOpenJournal of Inequalities and Applications1029-242X2017-09-012017111210.1186/s13660-017-1479-8Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequalityChao-Ping Chen0Hui-Jie Zhang1School of Mathematics and Informatics, Henan Polytechnic UniversitySchool of Mathematics and Informatics, Henan Polytechnic UniversityAbstract Based on the Padé approximation method, in this paper we determine the coefficients a j $a_{j}$ and b j $b_{j}$ ( 1 ≤ j ≤ k $1\leq j \leq k$ ) such that 1 e ( 1 + 1 x ) x = x k + a 1 x k − 1 + ⋯ + a k x k + b 1 x k − 1 + ⋯ + b k + O ( 1 x 2 k + 1 ) , x → ∞ , $$ \frac{1}{e} \biggl( 1+\frac{1}{x} \biggr) ^{x}= \frac{x^{k}+a_{1}x^{k-1}+ \cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots +b_{k}}+O \biggl( \frac{1}{x ^{2k+1}} \biggr) , \quad x\to \infty , $$ where k ≥ 1 $k\geq 1$ is any given integer. Based on the obtained result, we establish new upper bounds for ( 1 + 1 / x ) x $( 1+1/x ) ^{x}$ . As an application, we give a generalized Carleman-type inequality.http://link.springer.com/article/10.1186/s13660-017-1479-8Carleman’s inequalityweight coefficientPadé approximant |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Chao-Ping Chen Hui-Jie Zhang |
spellingShingle |
Chao-Ping Chen Hui-Jie Zhang Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality Journal of Inequalities and Applications Carleman’s inequality weight coefficient Padé approximant |
author_facet |
Chao-Ping Chen Hui-Jie Zhang |
author_sort |
Chao-Ping Chen |
title |
Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality |
title_short |
Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality |
title_full |
Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality |
title_fullStr |
Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality |
title_full_unstemmed |
Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality |
title_sort |
padé approximant related to inequalities involving the constant e and a generalized carleman-type inequality |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1029-242X |
publishDate |
2017-09-01 |
description |
Abstract Based on the Padé approximation method, in this paper we determine the coefficients a j $a_{j}$ and b j $b_{j}$ ( 1 ≤ j ≤ k $1\leq j \leq k$ ) such that 1 e ( 1 + 1 x ) x = x k + a 1 x k − 1 + ⋯ + a k x k + b 1 x k − 1 + ⋯ + b k + O ( 1 x 2 k + 1 ) , x → ∞ , $$ \frac{1}{e} \biggl( 1+\frac{1}{x} \biggr) ^{x}= \frac{x^{k}+a_{1}x^{k-1}+ \cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots +b_{k}}+O \biggl( \frac{1}{x ^{2k+1}} \biggr) , \quad x\to \infty , $$ where k ≥ 1 $k\geq 1$ is any given integer. Based on the obtained result, we establish new upper bounds for ( 1 + 1 / x ) x $( 1+1/x ) ^{x}$ . As an application, we give a generalized Carleman-type inequality. |
topic |
Carleman’s inequality weight coefficient Padé approximant |
url |
http://link.springer.com/article/10.1186/s13660-017-1479-8 |
work_keys_str_mv |
AT chaopingchen padeapproximantrelatedtoinequalitiesinvolvingtheconstanteandageneralizedcarlemantypeinequality AT huijiezhang padeapproximantrelatedtoinequalitiesinvolvingtheconstanteandageneralizedcarlemantypeinequality |
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1725233123870900224 |