I am wondering, since
$$ \frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z $$
can offer information about winding numbers by
$$ \frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z = \sum_{k=1}^{m} n(\gamma;a_k) $$
for $m$ is the max multiplicity of winding numbers, then, what about the following, are they giving anything about winding numbers or zeros of the function $f(z)$:
$$ \frac{1}{2 \pi i} \int_{\gamma} \frac{z f^{\prime}(z)}{f(z)} d z \quad \text { and } \quad \frac{1}{2 \pi i} \int_{\gamma} \frac{z^{2} f^{\prime}(z)}{f(z)} d z \quad \text { and } \quad \frac{1}{2 \pi i} \int_{\gamma} \frac{z^{3} f^{\prime}(z)}{f(z)} d z \quad \text { and } \quad \ldots $$