On Copson’s inequalities for 0 < p < 1 $0< p<1$ - Peng Gao; HuaYu Zhao (Journal of Inequalities and Applications, 2020)
Abstract Let ( λ n ) n ≥ 1 $(λ_{n})_{n ≥1}$ be a positive sequence and let Λ n = ∑ i = 1 n λ i $\varLambda_{n}=\sum^{n}_{i=1}λ_{i}$ . We study the following Copson inequality for 0 < p < 1 $0< p<1$ , L > p $L>p$ : ∑ n = 1 ∞ ( 1 Λ n ∑ k = n ∞ λ k x k ) p ≥ ( p L − p ) p ∑ n = 1 ∞ x n p . $$\begin{aligned} \sum^{∞}_{n=1} \Biggl(\frac{1}{\varLambda_{n}} \sum^{∞ }_{k=n}λ_{k} x_{k} \Biggr)^{p} ≥ \biggl( \frac{p}{L-p} \biggr)^{p} \sum^{∞}_{n=1}x^{p}_{n}. \end{aligned}$$ We find conditions on λ n $λ_{n}$ such that the above inequality is valid with the constant being the best possible.