Some extensions of Diananda's inequality

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 $\sum^n_{i=1}q_i=1$. 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 $\Delta_{1, 1/2, 0, 1}$ in terms of functions of $q$ only, where $q=\min q_i$. In this paper, we prove similar sharp bounds of $\Delta_{r,s,t,\alpha}$ for certain parameters $r, s, t, \alpha$.