${\left(\frac{z+1}{z}\right)}^4=16$

$\Rightarrow \frac{z+1}{z}=2$ or $-2$ or $2i$ or $-2i$
Hence, the roots are $1, -\frac{1}{3}, \left(-\frac{1}{5}-\frac{2}{5}i\right)$ and $\left(-\frac{1}{5}+\frac{2}{5}i\right)$

i.e. the roots are equidistant from the centre of the circle as shown in the figure.

$\left(\frac{1}{3},0\right)$ is the centre of the circle.

Hence, $h=\frac{1}{3}$