Suppose Suppose $A$ and $B$ are two entire, non-surjective, functions. This means


$$
A(z)=e^{f(z)}+c\_1
$$
and
$$
 B(z)=e^{g(z)}+c\_2
$$
for some entire functions $f$ and $g$, and some complex constants $c\_1$ and $c\_2$.


When is the sum $A(z)+B(z)$ an surjective entire function? More precisely, is there any condition on $f$ and $g$? Clearly the sum may fail to be surjective, for example when $f(z)=g(z)+c$. But also can be surjective, for example when $f(z)=2g(z)$. A naive conjecture would be that the sum is surjective as long as $f-g$ is not constant. Is that actually true?



Sums of non-surjective entire functions
This sum is surjective, with some trivial exceptions. Here is
a general Theorem of Emile Borel: Let $g\_j$ be entire, and
$$e^{g\_1}+\ldots+e^{g\_n}=0.$$
Then the set of indices $\{1,\ldots,n\}$ can be partitioned
into disjoint subsets $I\_k$, such that for $i,j\in I\_k$ we have $g\_i-g\_j$ is constant, and
$$\sum\_{j\in I\_k}e^{g\_j}=0,$$
for every $k$.


In your case, suppose that the sum of two $e^{g\_j}+c\_j$ is not surjective, then we have a relation of the form
$$e^{g\_1}+e^{g\_2}+e^{g\_3}=0,$$
or of the form
$$e^{g\_1}+e^{g\_2}+e^{g\_3}+e^c=0.$$
Applying Borel's theorem, we conclude in the first case that
$g\_i=g+c\_i,$ $1\leq i\leq 3$, $\sum\_{i=1}^3 e^{c\_i}=0$.
And in the second case, one of the $g\_i$ must be constant,
for example, $g\_3=-c$ and if $g\_1$ and $g\_2$ are not constants then their difference must be constant, namely $g\_1=g\_2=\pi i(2n+1)$.


A good reference for Borel's theorem is the book


Serge Lang, Introduction to complex hyperbolic spaces,
Springer, 1987.


