Refinements of Carleman's inequality (Q2740876)

By introducing two parameters \(\alpha\) and \(\beta\), and estimating the weight coefficient \(\omega(m)\) as NEWLINE\[NEWLINE\omega(m)=(1+\tfrac{1}{m})^m, \qquad m\in \mathbb{N},NEWLINE\]NEWLINE the author gives a refinement of Carleman's inequality asNEWLINE\[NEWLINE\sum_{n=1}^\infty (a_1a_2\cdots a_n)^{1/n} \leq e\sum_{m=1}^\infty \Big(1-\frac{\beta}{m}\Big)\Big(1+\frac{1}{m}\Big)^{-\alpha} a_m,NEWLINE\]NEWLINE with \(0\leq\alpha\leq\frac{1}{\ln 2}-1\), \(0\leq\beta\leq 1-\frac{2}{e},\) and \(e\beta+2^{1+\alpha} =e.\) NEWLINENEWLINENEWLINEReviewer's remark: It is a useful refinement of Carleman's inequality.