Is there an analytic function $f$ in the open unit disc such that $|f(z)|=e^{|z|}$ therein?

Question

Is there an analytic function $f$ in the open unit disc such that $|f(z)|=e^{|z|}$ therein?

My try:

Suppose such a function exists .Then $|f(0)|=e^0=1$ Also $|f|\ge 0$.

Now $e^{|z|}$ attains its minimum value i.e. $1$ at $0$. Thus $|f(z)|\ge 1$

I was trying to use the minimum modulus theorem which states that if $f$ is a non- constant analytic function such that $|f|$ attains its minimum value in an interior point of the unit disc then $f$ is constant.

Here $|f|$ attains its minimum value at $0$ which is an interior point of the unit disc and thus $f$ is constant which is false as $|f|=e^{|z|}$.

Thus no such analytic function exists.

Answer 1

This is a community wiki answer to notice that this question was answered in comments and remove the question from the unanswered list -- the answer is yes, the OP's proof is correct.