A function is given to be analytic in the closed disk:$|z|\leq 4$ with $Min|f(z)|=3$ on the circle $|z|=4$ and with $f(1)=2i.$
Can there be an example of such a function? I am having a hard time in coming up with such a function, as it violates the maximum modulus principle. Does such a function exist? Thanks beforehand.
Pick a unit disc automorphism st $g(1/4)=2i/3$ (a composition of $\frac{4z-1}{4-z}$ with $\frac{3z+2i}{3-2iz}$ will do as $1/4 \to 0 \to 2i/3$) and let $f(z)=3g(z/4)$. Clearly $|f(z)|=3, |z|=4$ and $f(1)=2i$
No violation of maximum modulus since $1/f$ is not analytic in the given disc as it has a zero somewhere.