Rouché's theorem versions With the conditions of theorem, any books put the condition $|f-g|<|g|$ (1) to conclude that $f$and $g$ has a same number of zeros into domain. But others books show that this condition can be reduce to $|f-g|<|f|+|g|$ (2).
Anybody has an example where the functions satisfies 2 but no 1?
 A: Another way to view such conditions is to divide by $g$:
$$
            \left|z-1\right| < \left|z\right|+1, \\
               \left|z-1\right| < 1.
$$
The first condition restricts $z$ to take values anywhere except on the negative real axis (including $0$.) The second condition restricts $z$ to take values in the open unit disk centered at $1$. A logarithm has a maximal domain described by $|z-1| < 1+|z|$; a holomorphic logarithm is defined on the second region as well, but the second region is nowhere near maximal in that regard.
To find an example that you want, make sure that $\frac{f}{g}$ traces out a path in the slitted plane, but not in the open unit disk centered at $1$. A simple example was provided in the comments: $f=iz$, $g=z$ because $f/g=i$ is outside the unit disk centered at $1$, but is inside the slitted plane where the negative real axis is removed from $\mathbb{C}$, provided either inequality holds on $C$.
In either case, the image of a closed curve under $f/g$ must have winding number $0$, which means that image of a closed curve $C$ under $f$ must have the same winding number as the image of $C$ under $g$.
