  Let $(\lambda_n)_{n \geq 1}$ be a non-negative sequence with $\lambda_1>0$
and let $\Lambda_n=\sum^n_{i=1}\lambda_i$. We study the following Copson
inequality for $0<p<1$, $L>p$, \begin{align*}
  \sum^{\infty}_{n=1}\left (\frac 1{\Lambda_n} \sum^{\infty}_{k=n}\lambda_k x_k
\right
  )^p \geq \left ( \frac {p}{L-p}\right )^p
  \sum^{\infty}_{n=1}x^p_n. \end{align*}
  We find conditions on $\lambda_n$ such that the above inequality is valid
with the constant being best possible.
