Suppose $f:\mathbb{C}\to\mathbb{C}$ is a complex function, and that $a_1,\cdots,a_k\in\mathbb{C}$ are complex roots of another function (it doesn't matter what it is for the purposes of this question). If it is the case that $$\sum_{i=1}^kf(a_i)$$is actually a real number, is it also the case that $$\sum_{i=1}^kf(a_i)^r$$ is also real for integer $r$?
Naturally it would make sense for this to be the case, but I cannot see a way to approach it. We could try something along the lines of showing that $$\textrm{Im}\left(\sum_{1\le i\le k}f(a_i)\right)=0\implies\textrm{Im}\left(\sum_{1\le i\le k}f(a_i)^r\right)=0$$ but it isn't obvious how to do so.