The discussion centers on the properties of holomorphic functions within the open unit disc, specifically addressing the assertion that a holomorphic function \( f(z) \) that takes real values only on the real axis can have at most one zero in the open disc. The argument principle and Rouche's theorem are referenced as key tools for analyzing the number of zeros. The conclusion drawn is that due to the nature of holomorphic functions and the constraints imposed by the real axis, non-real zeros are excluded, reinforcing the claim of a maximum of one zero within the unit disc.
PREREQUISITESStudents and researchers in complex analysis, mathematicians interested in the properties of holomorphic functions, and anyone studying the implications of the argument principle and Rouche's theorem.
hedipaldi said:Homework Statement
Let f(z) be holomorphic in the open unit disc taking real values only on the real axis.show that f has at most one zero in the open disc.
Homework Equations
Rouche's theorem,the argument principle.
The Attempt at a Solution
obviously,f does not have non-real zero's .I need help in counting the zero's by using the argument principle.