Notes on an inequality involving the constant \(e\) (Q1589728)

The main objective of this note is to prove that \[ \sum^{\infty}_{n=1} \lambda_{n+1} (a^{\lambda_1}_1 a^{\lambda_2}_2 \dots a_n^{\lambda_n})^{1/\Lambda_n} < e \sum^{\infty}_{n=1} [1+(\Lambda_n/\lambda_n+1/5)^{-1}]^{-1/2} \lambda_n a_n, \] 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 \(\sum^{\infty}_{n=1} \lambda_n a_n < \infty\). This result improves that of \textit{B. Yang} and \textit{L. Debnath} [J. Math. Anal. Appl. 223, No. 1, 347-353 (1998; Zbl 0910.26011)] and gives that of \textit{B. Yan} and \textit{G. Sun} [J. Math. Anal. Appl. 240, No. 1, 290-293 (1999; Zbl 0941.26012)] if \(\lambda_n = 1\) for all \(n\in\mathbb N\).