Let $A=\{z\in \mathbb{C}: a<|z|<b\}$. Suppose $f$ is continuous on the boundary of $A$. Suppose there is a sequence of holomorphic polynomials $\{p_n\}$ that converges uniformly to $f$ on $\{z\in \mathbb{C}: |z|=b\}$.
Prove or disprove the following:$\{p_n\}$ converges uniformly to $f$ on $\{z\in \mathbb{C}: |z|=a\}$
My Try:
The first thing that came to my mind was disproving, because there are functions that are continuous but not holomorphic. So, for example I took $f(z)=\bar{z}$ which is continuous but not holomorphic. However, I could not show that there is a sequence of holomorphic polynomials $\{p_n\}$ that converges uniformly to $f$ on $\{z\in \mathbb{C}: |z|=b\}$. No clue after that. Can somebody give me a hint? Also, am I on the right track?