1. The problem statement, all variables and given/known data Given that f is analytic in the open disc of radius one around zero, and that f(0)=0 and mod(f(z))<=1 for ever z in D(0;1). Define g(z) = f'(0) if z=0, f(z)/z elsewhere. I want to show g is analytic in D(0;1) 2. Relevant equations Basic stuff about analytic functions and series representations, etc. 3. The attempt at a solution Okay, so if z does not equal 0 then it is the quotient of two analytic functions (f is analytic by the hypothesis and any polynomial is analytic) hence g is analytic. If z=0 then I wrote f as a Taylor series: f(z) = f(0) + f'(0)z + (f''(0)/2)*z^2 +... so g'(0) = [f'(0)]' = ' = 0 thus g is differentiable at z=0; and I'm done. Is my reasoning correct?