Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for...
[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
Let f, g E H(C) be such that |f(z)| < \g(z)| for any z e C. Show that there exists a E D(0,1) such that f(z) = ag(z) for any z E C. (Hint: consider f/g and be careful with the zeros of g.)
4. Let Z ~ N(0,1) be a standard normal variable. Calculate the probability (a) P(1 <Z < 2). (b) P(-0.25 < < < 0.8). (c) P(Z = 0). (d) P(Z > -1).
5.72. Let A = A(0,1) and let g: A → be an analytic function sat- isfying 9(0) = 0 and 1g'(0) = 1 whose derivative is a bounded function in A. Show that w > (4m)-1 for every point w of C ~ g(A), where m = sup{]g'(x): z E A}; i.e., show that the range of g contains the disk A(0,(4m)–?). (Hint. Fix w belonging to C ~ g(A). Then w # 0. The function h defined by h(z)...
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]
Notation: In what follows, let D = {z:z<1} and H = {z: Im(2) >0}, 6. Let a CD be nonzero. Show that there is a unique automorphism f of D such that f(a) = 0 and f(0) = a. (Hint: Use Theorem 2.2, Chapter 8, Section 2.1 of Stein- Shakarchi (page 220.)
(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].
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C?
Exercise 3.38. Let the random variable Z have probability density function 24 fz(z) = -1 <z<1 otherwise. (a) Calculate E[Z]. (b) Calculate P(0 <Z<į). (c) Calculate P(Z < į 12 > 0). (d) Calculate all the moments E[Z"] for n= 1,2,3,... Your answer will be a formula that contains n.
Let f (2) be defined by: k-?, <<-1 f(3) = z? +, -1<x<1 - kr1 Which of the following values of k would make f (2) continuous on R? Ok=0 There is no such value for k Ok= -1 Ok= 1