Let \(\Omega\subseteq\CC\) be open, let \(f,g\) be meromorphic on \(\Omega\), and let \(\gamma\) be a positively oriented toy contour such that \(\gamma\) and its interior lie in \(\Omega\). Suppose that \(f\) and \(f+g\) have no zeros or poles on \(\gamma\), and that \[ \abs{g(z)}<\abs{f(z)}\qquad\text{for all }z\in\gamma. \] Let \(Z_h\) and \(P_h\) be the numbers of zeros and poles of \(h\) inside \(\gamma\), counted with multiplicity. Then \(\Index_{w=0}(f\circ\gamma)=\Index_{w=0}\big((f+g)\circ\gamma\big)\), and hence \[ Z_f-P_f=Z_{f+g}-P_{f+g}. \] In particular, if \(f\) and \(g\) are holomorphic on \(\Omega\), then \(f\) and \(f+g\) have the same number of zeros inside \(\gamma\).
On \(\gamma\), \(\frac{f+g}{f}=1+\frac gf\) takes values in the disc \(\ts{\abs{w-1}<1}\), which does not contain \(0\), so the closed curve \(\frac{f+g}{f}\circ\gamma\) has winding number \(0\) about \(0\). The logarithmic derivative of a product is the sum of the logarithmic derivatives, so by T-52HK6 \[ \Index_{w=0}\big((f+g)\circ\gamma\big)=\Index_{w=0}(f\circ\gamma)+\Index_{w=0}\Big(\frac{f+g}{f}\circ\gamma\Big)=\Index_{w=0}(f\circ\gamma). \] By the argument principle, \(\Index_{w=0}(h\circ\gamma)=Z_h-P_h\) for \(h=f\) and \(h=f+g\).