An analytic function having no element of negative x-axis in its range

Question

Let $f$ be an analytic function defined on open unit disc such that range of f is contained in $\Bbb{C}\setminus(-\infty,0]$.

Then I need to tell whether $f$ must be a constant function.

I was thinking of discarding this by an example. But I am not able to find one.

Please help.