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> 
> > 
> > Prove or disprove: there is an analytic function $f$ on 
> >  $D=\{z:~|z|<1\}$ such that $|f(z)|=e^{|z|},~z\in D.$ 
> > 
> > 
> > 
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**Attempt.** I believe that such a function does not exist (besides, a candidate function would be for example $e^z$, but $|e^z|=e^{Rez}\leq e^{|z|}$, although I can not provide a proof of my claim.

Thanks in advance!