1. The problem statement, all variables and given/known data Let f and g be two holomorphic functions in a connected open set D of the plane which have no zeros in D; if there is a sequence an of points such that lim an = a and an does not equal a for all n, and if f'(an)/f(an)=g'(an)/g(an) show that there is a constant c such that f=cg in D. 2. Relevant equations If f is identically zero at a point in a connected open set, then f is identically zero on the whole set. 3. The attempt at a solution I have shown that (f/g)'(a) = 0, but I don't see how that would imply that the derivative is identically zero at that point.