I would like to prove that this function is analytic. It is really hard for me to see.

We suppose that f is analytic and zero free in a domain D, the function I wish to prove is analytic is:

My professor tells me there is a standard argument where I take a and a small disk then take a fixed path from to and then a line segment from to , but I am not sure where to go from there.

Any help is greatly appreciated!