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

Abstract: 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.