A complement to Diananda's inequality

Let $M_{n,r}=({\sum }_{i=1}^n{q}_i{x}_i^r{)}^{\frac{1}{r}},r\mathrm{\ne }0$ and $M_{n,0}={\lim }_{r\to 0}{M}_{n,r}$ be the weighted power means of $n$ non-negative numbers $x_i$ with $q_i>0$ satisfying ${\sum }_{i=1}^n{q}_i=1$. 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.