Suppose that we have $\frac{\sum_{i=1}^{m}x_i\ln{w_i}}{\sum_{i=1}^{m}\ln{w_i}} < C$ with $x_i > 1$ and $\ln{w_i} > 1$ for all $1 \le i \le m$, $x_i$ being integers, and $w_i$ and $C$ being rational numbers. Is it possible for me to know the minimum and/or the maximum of $\frac{\sum_{i=1}^{m}x_i}{m}$?
Sorry if my question sounds like a stupid question.
Let $\ln w_{min}$ be the minimum of $\ln w_1, \ln w_2, \ldots \ln w_m$. Then, we have
$$ C \sum_{i=1}^{m}\ln{w_i} > \sum_{i=1}^{m}x_i\ln{w_i} \ge \ln w_{min}\sum_{i=1}^{m}x_i $$
$$ \frac{1}{m}\sum_{i=1}^{m}x_i < \frac{C}{m \ln w_{min}}\sum_{i=1}^{m}\ln{w_i} $$
If we do not know anything more about $w_i$ then this is the best we can do. But if we have an asymptotic or know the maximum value of $w_i$ then it will be possible to upper bound in terms of $C$ and $m$. Since $\ln w_i > 1$, we have the weaker inequality $$ \frac{1}{m}\sum_{i=1}^{m}x_i< \frac{C}{m}\sum_{i=1}^{m}\ln{w_i} $$
