This question is about Complex Analysis If fz is analytic in

This question is about \"Complex Analysis\"

If f(z) is analytic in the whole plane, show that the family formed by all functions f(kz) with constant k is normal in the extended sense in the annulus r_1

Solution

Given that f is entire function that is analytic on whole of complex plane

now suppose function f(Kz) is normal in extended sense on annulus given now claim is that we want f a polynomial , Given that f is entire function that is analytic on whole of complex plane so it has power series expansion that is it can be approximated by a polynomial function so it is polynomial

conversely if f is polynomial then we want f(Kz) to be normal on given annulus now given f is polynomial so it is entire i,e analytic on whole of complex plane   this implies that f is normal on whole complex plane so in particular on given annulus now as f(z) is entire so f(Kz) is entire and so by above argument for f (z) will be true for f(kz).