1.Pick out the true statements:
a. Let $f$ and $g$ be analytic in the disc $|z| < 2$ and let $f = g$ on the interval [$−1, 1$]. Then$ f ≡g$.
b. If $f$ is a non-constant polynomial with complex coefficients, then it can be factorized into (not necessarily distinct) linear factors.
c. There exists a non-constant analytic function in the disc $|z| < 1$ which assumes only real values.
2.Let $\omega⊂\mathbb{C}$ be an open and connected set and let $f : \omega →\mathbb{C} $ be an analytic function. Pick out the true statements:
a. f is bounded if $\omega$ is bounded.
b. f is bounded only if $\omega$ is bounded.
c. f is bounded if, and only if, $\omega$ is bounded
for the 1 st question
By fundamental theorem of algebra we can say that (b) is true. For (a) I am little confused that am I able to apply identity theorem or not. For (c) by applying Cauchy Riemann equation we get it is false. Are my approaches correct?
for the 2nd question
here I am little confused. I guess none of them is correct but not sure.can anyone provide me some counter examples