UMD Analysis Qualifying Exam/Aug12 Complex

Problem 2 Solution 2 Problem 4 Suppose is holomorphic on a region containing the disk and that if . How many solutions does the equation have in the disc ? Justify your answer. Solution 4 We know on . Similarly, since then on . This gives on . So by Rouché's theorem, since both functions are holomorphic (i.e. have no poles), then has the same number of zeros as on the domain . Since has only one zero (namely 0), then there is only one solution to inside the open disc . Observe that for any , since that would imply for some on the boundary, contradicting the hypothesis. Thus, there is only one solution to inside the open disc .