On Copson’s inequalities for $0< p<1$

Abstract

AbstractLet $(\lambda_{n})_{n \geq1}$(λn)n≥1 be a positive sequence and let $\varLambda_{n}=\sum^{n}_{i=1}\lambda_{i}$Λn=∑i=1nλi. We study the following Copson inequality for $0< p<1$0<p<1, $L>p$L>p: $$\begin{aligned} \sum^{\infty}_{n=1} \Biggl(\frac{1}{\varLambda_{n}} \sum^{\infty }_{k=n}\lambda_{k} x_{k} \Biggr)^{p} \geq \biggl( \frac{p}{L-p} \biggr)^{p} \sum^{\infty}_{n=1}x^{p}_{n}. \end{aligned}$$ ∑n=1∞(1Λn∑k=n∞λkxk)p≥(pL−p)p∑n=1∞xnp. We find conditions on $\lambda_{n}$λn such that the above inequality is valid with the constant being the best possible.