On Copson's inequalities for $0<p<1$

Let $({\lambda }_n{)}_{n\ge 1}$ be a non-negative sequence with ${\lambda }_1>0$ and let ${\mathrm{\Lambda }}_n={\sum }_{i=1}^n{\lambda }_i$. 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.