Recapturing a holomorphic function on an annulus from its mean boundary values

Let $D$ be an annulus in the complex plane with closure $\bar D$ and boundary $\partial D$. We prove that a function $f$, holomorphic in $D$ with ${C^{1 + \varepsilon }}(\partial D)$ boundary data for some $\varepsilon > 0$, is uniquely determined by its arithmetic means ${s_n}(f)$ and ${s_{0n}}(f)$ over equally spaced points on $\partial D$. We also give an explicit formula for recapturing $f$ from its means ${s_n}(f)$ and ${s_{0n}}(f)$. Furthermore, we derive the relations between ${s_n}(f)$ and ${s_{0n}}(f)$ which are necessary and sufficient for the analytic continuability of $f$ from $D$ to the whole disc.