Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality

Abstract

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.

Keywords

• Carleman’s inequality
• weight coefficient
• Padé approximant