In this paper, we prove that the discrete Copson inequality (E.T. Copson, \emph{ Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 49-51) of one-dimension in general cases admits an improvement. In fact we study the improvement of the following Copson's inequality \begin{ align*} &\displaystyle\sum_{ n=1}^{ \infty}\frac{ Q_{ n}^{ \alpha}|A_n-A_{ n-1}|^{ 2}}{ q_{ n}}\geq\frac{ (\alpha-1)^2}{ 4}\displaystyle\sum_{ n=1}^{ \infty} \frac{ q_{ n}}{ Q_{ n}^{ 2-\alpha}}|A_{ n}|^{ 2}, \end{ align*}where $\alpha\in[0,1)$, $A_{ n}=q_{ 1}a_{ 1}+ q_{ 2}a_{ 2}+ \ldots +q_{ n}a_{ n}$, $Q_{ n}=q_1+q_2+\ldots+q_{ n}$ for $n\in \mathbb{ N}$, $\{ q_n\}$ is a positive real sequence and $\{ a_n\}$ is a sequence of complex numbers. We show that if $\{ q_n\}$ is decreasing then the above inequality has an improvement for $\alpha\in [1/3, 1)$. We also prove that for some increasing sequences $\{ q_n\}$ the above inequality can also be improved. Indeed, we prove that for $q_{ n}=n$ and $q_n=n^3$, $n\in \mathbb{ N}$ the corresponding Copson inequalities admit an improvement for $\alpha\in[\frac{ 17}{ 50}, 1)$ and $\alpha\in[0, \frac{ 1}{ 2}]$, respectively. Further, we show that in case of $q_{ n}=1$, $n\in \mathbb{ N}$ the reduced Copson inequality (known as Hardy's inequality with power weights) has achieved an improvement for $\alpha\in[0, 1)$.
Categories: math.CA math.FA