A complement to Diananda's inequality

TL;DR

This paper extends Diananda's inequalities for weighted power means by establishing their reversed forms, providing new bounds and insights into the relationships between different means of non-negative numbers.

Contribution

It introduces the reversed inequalities analogous to Diananda's original results, broadening the understanding of mean inequalities for weighted power means.

Findings

Reversed inequalities for weighted power means are proven.

New bounds for the difference between means are established.

The results complement existing inequalities by Diananda.

Abstract

Let $M_{n,r}=(\sum_{i=1}^{n}q_ix_i^r)^{\frac {1}{r}}, r \neq 0$ and $M_{n,0}=\lim_{r \rightarrow 0}M_{n,r}$ be the weighted power means of $n$ non-negative numbers $x_i$ with $q_i > 0$ satisfying $\sum^n_{i=1}q_i=1$. In particular, $A_n=M_{n,1}, G_n=M_{n,0}$ are the arithmetic and geometric means of these numbers, respectively. A result of Diananda shows that \begin{align*} M_{n,1/2}-qA_n-(1-q)G_n & \geq 0,\\ M_{n,1/2}-(1-q)A_n-qG_n & \leq 0,\end{align*} where $q=\min q_i$. In this paper, we prove analogue inequalities in the reversed direction.