Refinements of Carleman's inequality and its generalizations (Q1860733)

The author exploits the properties of power mean inequality with increasing weight coefficients to obtain some refinements of the Carleman's inequality and some of its consequences. For the sake of completeness, we reproduce the following as one of the major results obtained in this paper: If \(0 < \lambda_{n+1} \leq \lambda_n\), \(\Lambda_n = \sum^n_{m=1}\lambda_m\) \((\Lambda_n \geq 1\), \(a_n \geq 0)\), \(0< p \leq 1\) and \(0 < \sum^{\infty}_{n=1} \lambda_n a_n < \infty\), then \[ \sum^{\infty}_{n=1}\lambda_{n+1}\Bigl(a_1^{\lambda_1}, a_2^{\lambda_2},\dots, a_n^{\lambda_n}\Bigr)^{\frac 1 {\Lambda_n}} < \frac{e^p}{p}\sum^{\infty}_{n=1}\biggl(\frac {\Lambda_n^{\alpha}(\Lambda_n -\lambda_n \beta)} {\Lambda_n(\Lambda_n +\lambda_n)^{\alpha}}\biggr)^p\lambda_n a_n^p \Lambda^{p-1}_n\Biggl(\sum^n_{k=1}\lambda_k(c_k a_k)^p \Biggr)^{\frac{1-p}{p}}, \] where \(c_k^{\lambda_k} = \frac{\Lambda_{k+1}}{\Lambda_k^{\Lambda _{k-1}}}\), \(0\leq \alpha\leq 1/\ln 2-1\), \(0\leq \beta \leq 1-\frac{2}{e}\) and \(e\beta + 2^{1+\alpha} = e\).