let f be an entire function such that $|f(z) + e^z| > |e^zf(z)|$ for all z in C.
Show that f is a constant function.
Suggestion is to use Liouville's theorem then show that f is actually the constant zero function. But don't even know how to start? I'm sure I need to form a g(z) but not sure what my g should be.
Best Answer
$g(z) = (e^zf(z))/(f(z) + e^z)$ and we know $|g(z)| < 1$, entire and bounded implies constant. From it try to deduce this constant is 0 and which will imply in eventuality that $f(z) \equiv 0$.