I have the following optimization problem:

$$\text{find } x= \min_{h,a} \max_{\lambda\in\Lambda} |R(h\lambda)|$$

where $\Lambda$ is some finite, fixed set of complex numbers and $R(z)$ is a polynomial of the form

$$R(z) = 1 + z + \sum_{j=2}^{s}a_j z^j$$

where $h>0$ and the real coefficients $a_j$ are the decision variables. Can it be shown that $x$ is a differentiable function of $h$? Are there general tools that might be useful for proving this type of property?