I'm having trouble with part of the following problem:
Suppose is an open set in the complex plane , and let be a point of . Let be a holomorphic function on , and suppose .
First I'm suppose to show that there exists an , such that for all , the ball of radius and center .
This follows from continuity of at , with epsilon .
Then I'm showing that for all , it holds that
This follows from the calculation
with the last inequality following from the fact that the line segment between and is contained in .
Then I'm asked to prove that for all and in , it is true that
which I do by substituting a single variable for the expression in the integral.
My problem is to show that this implies that restricted to the ball is injective, which boils down to showing that the integral is non-zero for . Any ideas?