[3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that [f'(x) =...
[3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that \f'(x) < 1/(1 - 1z| for all z e D[0, 1]. [3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that f'(x) < 1/(1-1-12 for all z e D[0, 1]
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2) < 1/(1 - 121) for all z e D[0,1].
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2
.Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that |f 0 (z)| ≤ 1/(1 − |z|) ^2 for all z ∈ D[0, 1]. is there any way not to use the Schwartz' lemma
[3] 5. Suppose that f: D[0,1] for all z E D(0,1) D[0,1] is holomorphic, prove that f'() 5 1/(1 - 121)?
help with thus problem but not using schwoz-pick lemma [3] 5. Suppose that f: D[0,1] → D[0,1] is holomorphic, prove that f'(x) < 1/(1 - 1-1) for all z € D[0, 1]
Please don't use schwarz pick lemma 5.17. Suppose f : D[0,1] → D[0,1] is holomorphic. Prove that for z1 <1, 1 |f'(2) 1 - 12
Suppose f(z) is a holomorphic function in a domain U, and z0 ∈ U. Prove that f has a zero of order m at z0 if and only if f(z) = g(z)(z − z0)^m, where g(z) is holomorphic in U and g(z0) not equal to 0. Please prove both directions of the if and only if statement and use series expansion to prove. We have not learned calculus of residues yet.
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...