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

TL;DR

This paper investigates the validity of a specific Copson inequality for sequences with parameters $0<p<1$ and $L>p$, establishing conditions on the sequence for the inequality to hold with the optimal constant.

Contribution

It provides new conditions on the sequence $(\lambda_n)$ ensuring the inequality's validity with the best possible constant for the case $0<p<1$.

Findings

Derived necessary and sufficient conditions on $(\lambda_n)$ for the inequality to hold.

Identified the best possible constant in the inequality.

Extended the understanding of Copson inequalities in the range $0<p<1$.

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.