Entire function with positive real part is constant (no Picard) [duplicate]
Solution 1:
Write $f = u + iv$, where by assumption $u \geq 0$. Consider the analytic function
$$\exp(-f), \,\,\,\,\,\,\, |\exp(-f)| = |\exp(-u)|$$
What can we say about $\exp(-f)$? (Hint: Louiville's Theorem)
Solution 2:
Compose your map with $e^{-z}$, and you get a bounded entire function.
Solution 3:
Hint: a fractional linear transformation (aka Möbius transformation) takes a half-plane to a disk.