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Guys, I need help with this question. I honestly don't even my take on this one yet because I do not know where or how to start. Please help/direct me.

Thanks in advance

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Have you seen Green's theorem? This is Green's theorem rephrased in terms of complex line integrals and the "conjugate derivative" $\partial_{\overline{z}}$. Write $f$ as a function of $x$ and $y$, and convert everything to real integrals.

Well, actually, that should be $\frac1{2i}\oint_{\gamma} f(z)\,dz$ on the left. You can verify this yourself with the example $f(z)=\overline{z}$ on the unit circle.

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