Complex Analysis Question (Hadamard-related?)

The following is an (old) exam question I am unsure of:

:Construct an entire f(z) such that lim r--> infinity (log M(r)/r^.5)
=1 where
M(r)=max over /z/=r of /f(z)/. Further, does every f(z) with this
growth rate have infinitely many zeros?:

The second question made me doubt my approach to the first question.
How do you resolve these questions? Any help would be appreciated.


Thank you.

.

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