It is curious when one will finally achieve a moment of insight.

Only after submitting my post a moment ago and reviewing it for typographical error, did the solution come to mind.

I have established that f maps C to the right half plane. I can now map the right half plane to the unit disk by a Mobius transformation, say g(z) = (1-z)/(1+z). The composition map gf is entire, and bounded by 1, so gf is constant by Liouville's Theorem. Supposing gf(z) = C,

[1 - f(z)] / [1 + f(z)] = C implies f(z) = (1-c)/(1+c), for which u being constant is requisite.

I am glad I was able to do this myself, but thank you anyway.

(As an interesting corollary, there must be no entire function reducing to a half plane)