$f$ analytic on $\mathbb{C}-\left\{0\right\},$ $\text{Im}f<-1$ for $|z| = 1/2,2$; show $f(1)\neq 0$.

Question

Suppose $f$ is analytic on $\mathbb{C}-\left\{0\right\}$ and that $ \Im(f) < -1$ for $|z|=1/2$ and $|z| = 2$. Show $f(1)\neq 0.$

I have tried Cauchy's Integral formula, but it seemed to be a dead end. Any and all hints are appreciated.