Improved Carleman's Inequality and Improvement of Polya-Knopp's Inequality

This note, we first show that the famous Carleman's inequality can be improved if we find a positive sequence $\{c_n\}$ such that $c_n{\sum\limits_{j=n}^{\infty}}{\frac{1}{j\(\prod_{k=1}^{j}ck\)^{\frac{1}{j}}}}$ < e. Then we list a lot of known results in the literature improving Carleman's inequality by this method. These results can be a good source to a further research for interested students. We next consider about similar improvement of Polya-Knopp's inequality, which is a continuous version of Carleman's inequality. We show by a manner parallel to the case of Carleman's inequality that Polya-Knopp's inequality can be improved if we find a positive function c(x) such that $c(x){\int}_{x}^{\infty}\frac{1}{t\;{\exp}\(\frac{1}{t}{\int}_{0}^{t}{\ln}\;c(s)\;ds\)}dt$ < e. But there are no known results improving Polya-Knopp's inequality by this method. Suggesting to find a new method, we lastly show that there is no nice continuous function c(x) that satisfies the inequality.

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