On a result of Cartwright and Field

Let $M_{n,r}=(\sum_{i=1}^{n}q_ix_i^r)^{\frac {1}{r}}, r\neq 0$ and $M_{n,0}=\displaystyle \lim_{r \rightarrow 0}M_{n,r}$ be the weighted power means of $n$ non-negative numbers $x_i, 1 \leq i \leq n$ with $q_i > 0$ satisfying $\sum^n_{i=1}q_i=1$. Let $r>s$, a result of Cartwright and Field shows that when $r=1, s=0$, \begin{align*} \frac {r-s}{2x_n}\sigma_n \leq M_{n,r}-M_{n,s} \leq \frac {r-s}{2x_1} \sigma_n, \end{align*} where $x_1=\min \{x_i \}, x_n=\max \{x_i \}, \sigma_n=\sum_{i=1}^{n}q_i(x_i-M_{n,1})^2$. In this paper, we determine all the pairs $(r,s)$ such that the right-hand side inequality above holds and all the pairs $(r,s), -1/2 \leq s \leq 1$ such that the left-hand side inequality above holds.