A best approximation for constant \(e\) and an improvement to Hardy's inequality
The authors show that \[ e\Big(1-\frac 7{14x+12}\Big) < \Big(1+\frac 1{x}\Big)^x < e \Big(1-\frac 6{12x+11} \Big), \quad x \geq 1, \] and apply this result to prove that \[ \sum^{\infty}_{n=1} \lambda_{n+1} (a^{\lambda_1}_1 a^{\lambda_2}_2 \dots a^{\lambda_n}_n)^{1/\Lambda_n} < e \sum^{\infty}_{n=1} \Big(1-\frac{6\lambda_n}{12\Lambda_n + 11 \lambda_n}\Big) \lambda_n a_n, \leqno(\ast) \] where \(0< \lambda_{n+1} \leq \lambda_n\), \(\Lambda_n = \sum^n_{m=1} \lambda_m\), \(a_n\geq 0\) \((n\in\mathbb N)\), and \(0<\sum^{\infty}_{n=1} \lambda_n a_n < \infty\). Note that \((*)\) yields an improvement of the well-known Carleman inequality.
scientific article; zbMATH DE number 1563107