If $\abs{f-g}<\abs{f}+\abs{g}$ on the boundary circle, then $f$ and $g$ have the same number of zeros in the disc.Suggest a slogan
Let \(D=\theset{z : \abs{z-z_0}<R}\), and let \(f,g\) be holomorphic on an open set containing \(\overline D\). If \(\abs{f(z) - g(z)} < \abs{f(z)} + \abs{g(z)}\) for all \(z\in\bd D\), then \(f\) and \(g\) are nonvanishing on \(\bd D\) and have the same number of zeros in \(D\), counted with multiplicity.
What does Rouche's theorem conclude about the zero counts of $f$ and $g$?