The polylogarithm function (aka Jonquière's function) is defined as $Li_s(z) = \sum_{k=1}^{\infty} z^k k^{-s}$.
Is there a closed-form upper bound for this function when $s \leq -1$ and is real, and $z \in [0, 1]$ and is also real?
Note: this question is similar but the OP was looking for a constant bound, which is not possible. I am looking for an upper bound as a function of $z$ and $s$.
I don't know how good the upper bound needs to be, but one approach would be to start by noticing that if $s\leq t$ then $$ z^kk^{-s} \geq z^kk^{-t}$$ since $k\geq 1$. This means $$ Li_s(z) \geq Li_t(z),$$ hence in particular $$ Li_s(z) \leq Li_{\lfloor s\rfloor}(z)$$ since $\lfloor s\rfloor \leq s$. But since $$Li_0(z) = \frac{z}{1-z}$$ and $$ \frac{\partial Li_s(z)}{\partial z} = Li_{s-1}(z)$$ we find for $s<0$ that $$ Li_s(z) \leq Li_{\lfloor s\rfloor}(z) = \Big(\frac{\partial}{\partial z}\Big)^{\lfloor s\rfloor} \frac{z}{1-z}.$$
