Suppose that f : D - C is holomorphic with 2 |f' (0) | = sup If(z) - f(w)I Z,WEDD Prove that f is linear. My attempt Suppose that f(z) = > "anz" n=0 Then 2 If' (0) | = 2|all, and 2, WED sup If(z) - f(w)| > sup If(z) - f(-z)1 | |=1 = 2|a1 | sup 1 + a3z2 + asz* + ...| 2|=1 > 2 a1 where we use the maximum modulus principle. But I got stuck to show that the equality holds precisely when a2 = a3 = . .. = 0. Any hints would be highly appreciated