JIPAM - Journal of Inequalities in Pure and Applied Mathematics

Volume 4, Issue 1, Article 2

Note on the Carleman's Inequality for a Negative Power Number

Authors:

Thanh Long Nguyen, Vu Duy Linh Nguyen, Thi Thu Van Nguyen,

Keywords:

Carleman's inequality.

Date Received:

30/10/02

Date Accepted:

25/11/02

Subject Codes:

26D15

Editors:

Hüseyin Bor,

Abstract:

By the method of indeterminate coefficients we prove the inequality

$displaystyle sum_{n=1}^{infty }frac{n}{frac{1}{a_{1}}+frac{1}{a {2}}+...+frac{1}{%% a_{n}}}$

$displaystyle leq 2sum_{n=1}^{infty }left( 1-frac{1}{3n+1}%% -4n^{2}sum_{k=n}^{infty }frac{1}{k(k+1)^{2}(3k+1)(3k+4)}right) a_{n},$

where $a_{n}>0, n=1,2,... sum_{n=1}^{infty }a_{n}<infty .$


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