Refinements of Carleman's inequality and its generalizations
type: Item
instance of: scholarly article
title: Refinements of Carleman's inequality and its generalizations
zbMATH Open document ID: 1026.26013
author: Young-Ho Kim
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2003-11-30T00:00:00Z
review text: 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 \(e\beta + 2^{1+\alpha} = e\).
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1874463
zbMATH Keywords: Carleman's inequality
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: monotonicity
zbMATH Keywords: power mean inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1874463