Jc 22 - 22 terclockwise. Problem 38. Prove that if f, g are in LP(0,1) with...
[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
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
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny
Problem 2: Solve the initial value problem: with 4. 0<t〈2 f(t) 14t-2i,22
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
Let U ? Rmxn. Prove that if UTI-In, then n < m.
(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].
Let ne Nj. Prove that n < 2(6(n)).