Show that there is no holomorphic fuction $f$ in the unit disc $D$ that extends continuously to boundary of $D$ such that $f(z)=\frac{1}{z} ~for~ z\in \partial( D) $.
I tried to apply maximum principle but I couln't find the way to prove it.
Help me please.
I just update the full statement and I think it probably assume it is not constant fuction.
Thank you.
Best Answer
If such a function existed, then for all $r < 1$ you'd have $\int_{|z| = r} f(z)\,dz = 0$ by Cauchy's Theorem. Taking limits as $r \rightarrow 1$ would give $\int_{|z| = 1} f(z)\,dz = 0$ as well, contradicting that $\int_{|z| = 1} {1 \over z}\,dz = 2\pi i$.