Part (a): The function f is analytic in the whole plane with positive imaginary part. What can it be?
Part (b): What if all you know is that the imaginary part of f tends to 0 at infinity?
what we know:
For part(a): Write f(z) = u(x,y) + iv(x,y)
u(x,y) and v(x,y) are both harmonic; in particular, they are harmonic conjugates to each other. So, Uxx+Uyy=0, and similarly for v.
The Cauchy-Riemann equations hold: $$Ux = Vy$$ $$Uy=-Vx$$
Since f has positive imaginary part, then v(x,y) $>/=$ 0, and its partial derivatives, Vx and Vy are both non-negative.
What more can I say about this function, for part(a)?
Some natural (or maybe not so natural?) guesses would be that f is a constant, or a polynomial, or an exponential function, since we know that f is entire. But, at the moment, I can't seem to extract any more information from the question itself.
Thanks in advance,