Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$ ,satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin and set $\partial D := c$. Assume that $F$ has no zeros in $\partial D$.
(a) Prove that the index $ n(F; D)$ is an odd integer.
(b) Prove that $F$ has at least one zero on the disk $D$.
Definition: $\displaystyle{ n(F;D) =\frac{1}{2 \pi} \int_c \theta _0}$ where $\displaystyle{ \theta_0 = \frac{g df - fdg}{f^2 + g^2}}$
One can easily check that the index $n(F;D)$ is the winding number of the curve $ F \circ c$ about the origin and so index is an integer.
edit: I didn't write all hypothesis. I am really sorry for that. I hope now is clear.