Hey there, I realy need some guidance in the following questions:
1. Let f,g be entire functions and there exists a real constant M such as Re(f(z)) <= M * Re(g(z)) for every z in C.
Prove that there exist complex numbers a,b such as f(z) = a*g(z) +b for every z in C.
2. Let f(z) be analytic at the open unit circle and
|f ' (z) | <= 1/ (1 - |z| )for every |z| < 1.
Prove that the coefficients in that taylor series f(z) = Sigma_ an*zn are : |an| < e for every n>=1.
About 1- If we'll define h(z) = f(z)/g(z) we'll get a constant function by Liouville's theorem. But I can't understand how to get the "b" in the expression...
About 2-I'm pretty sure we need to use Cauchy's Inequality but I can't figure out how...
I will be delighted to get some guidance around here...
Thanks a lot