Entire functions such that $\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty$ The problem I am working on is to find all entire functions satisfying $|f(z)| > 0$ for $|z|$ large and $$\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty.$$
My guess is that these functions are the entire functions of order 1? Outside a sufficiently large ball centered at 0, $\log f$ is holomorphic and hence the numerator is really just $|\textrm{Re} \log f|$.
 A: 
Outside a sufficiently large ball centered at $0$, $\log f$ is holomorphic 

Not really, unless your definition of "holomorphic" allows multi-valued functions, which would be unusual. For example $\log z$  does not have a holomorphic branch in $|z|>1$.
I think what you should do is: 


*

*Observe that $f$ has finitely many zeros.

*Divide $f$ by a polynomial $p$ to get rid of zeros.

*Take logarithm (now we know it's holomorphic, because the domain is simply connected, so the Monodromy theorem applies). 

*Argue that $\log (f/p)$ is a polynomial of pretty small degree. 


Tangential remarks are below the cut.

Such functions have order $1$ or less. Also, not every function of order $1$ qualifies; it must have finite type. The standard way to describe such functions is to say thet are of exponential type. 
Many (but not all)   functions of exponential type arise as Fourier transforms (the Paley-Wiener theorem). A complete characterization is provided by the Hadamard factorization theorem, which for such functions takes the simple form
$$
f(z) = Ce^{bz} z^k \prod_{n} \left( 1 - \frac z{z_n} \right) \exp\left( \frac z{z_n} \right)
$$
where $z_n$ are the nonzero zeros of $f$ ordered by magnitude and $k$ is the multiplicity of zero at zero. :)
When you have only finitely many zeros,   the above formula   simplifies.  
