1. The problem statement, all variables and given/known data Suppose that f is an entire function. Define g(z)=f*(z*), where * indicates conjugates. I know from another problem that g(z) is also entire. Suppose also that f(z) maps the real axis into the real axis, so that f(x+0i)is in R for at x in R. Show that f(z)=g(z) for all z in C. 2. Relevant equations There is a hint that says, consider f(z)-g(z). I know that if the derivative of something is 0, then it is constant, and if we know that value at one point we know it is that value everywhere. 3. The attempt at a solution I know that on the real axis, f(z)=f(z*)=f*(z*), so that f(z)-g(z)=0 for all z such that z=x+0i. I was hoping to either show that the derivative is constant, or perhaps use something about isolated zeros? Is it true that a non-constant entire function has isolated zeros?