Let $\displaystyle f$ be analytic in the open disk $\displaystyle D(z_{0},R)$. Let $\displaystyle r< R\hat$ and $\displaystyle \gamma_{r}$ be the anticlockwise oreinted circle of radius $\displaystyle r$ centered at $\displaystyle z_{0}$.

Let $\displaystyle w \in$ tr$\displaystyle \gamma_{r}$.

Does it follow that $\displaystyle f(w)$ is bounded on tr$\displaystyle \gamma_{r}$?

Thanks