TITLE: There are no $f,g:\mathbb{C}\to\mathbb{C}$, entire end $e^{f(z)}=e^{g(z)}+c$.
QUESTION [0 upvotes]: I am trying to prove Picard's little theorem. Here are my steps:

If $a,b\notin Im(\phi(\mathbb{C}))$ then $\phi_1(z)=\phi(z)-a$ and $\phi_2(z)=\phi(z)-b$ don't vanishes any where. then there are entire functions $f,g:\mathbb{C}\to\mathbb{C}$ such that $\phi_1(z)=e^{f(z)}, \phi_2(z)=e^{g(z)}$. Then we have: $$e^{f(z)}+a=e^{g(z)}+b$$

Which gives title function equation.

REPLY [0 votes]: I suggest you take a look at Rudin, Real and Complex Analysis, Theorem 13.11.
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