Some inequalities involving the constant \(e\), and an application to Carleman's inequality (Q1269610)

The authors prove that for all \(n\in\mathbb{N}\), \[ e\Biggl[1- {1\over 2(n+5/6)}\Biggr]< \Biggl(1+{1\over n}\Biggr)^n< e\Biggl[1- {1\over 2(n+1)}\Biggr] \] and apply this result to strengthen the well-known Carleman's inequality \[ \sum^\infty_{n=1} (a_1a_2\cdots a_n)^{1/n}< e \sum^\infty_{n=1} a_n,\tag{\(*\)} \] where \(a_n\geq 0\) and \(0< \sum^\infty_{n= 1}a_n< \infty\). They prove that the sum appearing on the right-hand side of \((*)\) can be replaced by \(\sum^\infty_{n=1} \left[1-{1\over 2(n+ 1)}\right] a_n\).