For a given real finite series, say $\{x_n\}$, and a subset of this series $\{y_m\}$ ($m < n$), that is a subsequent serie where some terms of $\{x_n\}$ have been withdrawn. If we would to compare and bound difference between series means, is there a tighter inequality than:
$$ |\bar{x} - \bar{y}| \leq \max(\{x_n\}) - \min(\{x_n\}) $$
Or is it the only thing we can say about it.
The inequality can be slightly strengthened to: $$|\bar{y}-\bar{x}|\leq \left(1-\frac{m}{n}\right)(\max\{x_n\}-\min\{x_n\})$$ Suppose $S\subset \{1, \cdots, n\}$ is the size $m$ set of indices corresponding to the $y_i$. Then note that: \begin{align*} n|\bar{x}-\bar{y}|&=\left|\sum_i x_i-\sum_{i\in S}\frac{n}{m}x_i\right| \\ &=\left|\sum_{i\notin S}x_i-\frac{n-m}{m}\sum_{i\in S}x_i\right| \\ &=\left|\sum_{i\notin S}(x_i-\bar{y})\right| \\ &\le \sum_{i\notin S}\left|x_i-\bar{y}\right| \\ &\le (n-m)(\max\{x_n\}-\min\{x_n\})\end{align*} and dividing by $n$ gives the required result. This is sharp when the $x_i$ take on just two values, and all instances of one of the two values are selected for the subsequence $y_i$.