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.
Categories: math.CA