Some extensions of 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 ${\sum }_{i=1}^n{q}_i=1$. For a real number $\alpha$ and mutually distinct real numbers $r,s,t$, we define \begin{align*} \Delta_{r,s,t,\alpha}=\Big | \frac {M^{\alpha}{n,r}-M^{\alpha}{n,t}}{M^{\alpha}{n,r}-M^{\alpha}{n,s}}\Big |. \end{align*} A result of Diananda gives sharp bounds of ${\mathrm{\Delta }}_{1,1\mathrm{/}2,0,1}$ in terms of functions of $q$ only, where $q=\min q_i$. In this paper, we prove similar sharp bounds of ${\mathrm{\Delta }}_{r,s,t,\alpha }$ for certain parameters $r, s, t, \alpha$.