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

Abstract

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

Keywords

• Copson’s inequalities