Let f be an analytic function defined on the unit disc D={z in C:mod(z)<1}.if mod(f(z))</=1-mod(z) for each

let f be an analytic function defined on the unit disc D={z in C:mod(z)%26lt;1}.if mod(f(z))%26lt;/=1-mod(z) for each z in D,show that f is the zero function on D

Let f be an analytic function defined on the unit disc D={z in C:mod(z)%26lt;1}.if mod(f(z))%26lt;/=1-mod(z) for each
Fix z in D. By the maximum modulus theorem for analytic functions, for r with |z| %26lt; r %26lt; 1, |f| either has no local maximum in { w : |w| %26lt; r } or is constant on that domain. Thus





|f(z)| =%26lt; sup { |f(w)| : w in C with |w| = r } =%26lt; 1 - r.





Since the r chosen was arbitrary, it follows that |f(z)| = 0, from which we deduce that f is zero on D.