Is there a non-Shih analog for holomorphic functions of the Intermediate Value Theorem? Let $C$ be a simple closed curve in the complex plane, and let $f$ be holomorphic on an open set containing $C$. Is there a condition on the signs of Im $f$ and Re $f$ on $C$ that guarantees the existence of a zero of $f$ in the interior of $C$? (Shih's theorem is not quite what I want; there the condition is   $\text{Re }\overline{z} \cdot f(z) > 0$.) (See M.-H. Shih, "Bolzano's Theorem for Functions of a complex variable, Am. Math. Monthly v. 89 (1982), 210-211.) 
If the number of zeros of $f$ inside $C =$ the number of zeros of the identity in $f(C)$ then it should be sufficient that $f(C)$ passes through each quadrant in cyclic order (which can be rewritten as conditions on the imaginary and real parts as originally requested.) When is this equality the case?
 A: "The number of zeros of identity" is the strange expression: this number is always $0$ or $1$. Assuming that $f$ is analytic in an open set containing $C$ and its interior
region, and has no zeros on $C$, the number of zeros inside $C$ is always equal
to the index of the curve $f(C)$ about $0$,
$$\frac{1}{2\pi i}\int_{f(C)}\frac{dw}{w}=\frac{1}{2\pi i}\int_C\frac{f'(z)}{f(z)}dz.$$
All other counting formulas for zeros follow from this, as Robert Israel wrote.
This can be counted in terms of the numbers of crossing of the coordinate axes by $u=\Re f$
and $v=\Im f$. For example, consider those points on $C$ where $u(z)>0$ and $v$ changes sign
at $z$ (as $z$ moves on $C$ in positive direction). Then the index equals to the 
number of changes from $-$ to $+$ minus the number of changes from $+$ to $-$ at these points. 
A: I don't know what kind of condition you're looking for, but whatever condition it is must essentially work through the Argument Principle: as $z$ goes around the (positively oriented) curve $C$, it must force $f(z)$ to have winding number $\ge 1$ around $0$.  
