Any entire analytic function whose range omits two points must be a constant function.
Of course, an entire function that omits a single point from its range need not be a constant, as illustrated by the function , which is entire but omits the point from its range.
More things to try:
Weisstein, Eric W. "Picard's Little Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PicardsLittleTheorem.html