Abstract
Based on the Padé approximation method, in this paper we determine the coefficients a_{j} and b_{j} (1leq j leq k) such that \t\t\t1e(1+1x)x=xk+a1xk−1+⋯+akxk+b1xk−1+⋯+bk+O(1x2k+1),x→∞,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\frac{1}{e} \\biggl( 1+\\frac{1}{x} \\biggr) ^{x}= \\frac{x^{k}+a_{1}x^{k-1}+ \\cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\\cdots +b_{k}}+O \\biggl( \\frac{1}{x ^{2k+1}} \\biggr) , \\quad x\\to \\infty , $$\\end{document} where kgeq 1 is any given integer. Based on the obtained result, we establish new upper bounds for ( 1+1/x ) ^{x}. As an application, we give a generalized Carleman-type inequality.
Highlights
Let an ≥ for n ∈ N := {, . . .} and < ∞ n= an < ∞
The inequality ( . ) was presented in in [ ] by Carleman and it is called Carleman’s inequality. Carleman discovered this inequality during his important work on quasi-analytical functions
Following the same method used in the proof of Theorem . , we can prove the inequality ( . )
Summary
Based on the Padé approximation method, in this paper we determine the coefficients aj and bj (1 ≤ j ≤ k) such that ) was presented in in [ ] by Carleman and it is called Carleman’s inequality. Carleman discovered this inequality during his important work on quasi-analytical functions. ) have been given by estimating the weight coefficient ( + /n)n. Used it to obtain the following strengthened Carleman inequality:
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