An improved Copson inequality

Abstract

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 $\alpha\in[0, 1)$.