Complex Analysis math

Problem 2. Let f, h E H (C), and for every 2 E (C, we have |f(2)| |9(Z)| Show that f (z) = c g (z) for some constant c E (C. Problem 3. Let f, h E H (C), and Ref Sic Reg, for some EC Show that there exists a, b E C such that f (z) = a g (2:) + b. Problem 4. f is complex valued continuous function on [0,1], and 1 g(z)=/Df(t)etzdt, VZEC Show that g E H (C). Problem 5. Let f E CO ((C) be holomorphic on {2: : Rez 79 0}. Show that f is entire. Problem 6. Suppose \"um(um, = 0 for some positive function u on R2. Show that u is constant. Problem 7. Show that P (z) = 7,47 Z23 + 22:11 215 + 42:2 + 1 has at least one zero in 1D)