Answer to Question #125907 in Complex Analysis for Sanjana

Question

Determine whether the statement is true or false. Justify the answer.
If f is analytic in a convex domain D such that Re f'(z) is not equal to 0 for all z belongs to D, then f is univalent in D

Answer 1

Given statement is true.

Since, $f'(z)\neq 0 \ \forall z\in D$

Let $z_1, z_2\in D$ and $z_1\neq z_2$

Now, since $f'(z)\neq 0 \implies \frac{f(z_1)-f(z_2)}{z_1-z_2} \neq 0$ .

As $z_1\neq z_2 \implies f(z_1)\neq f(z_2)$

Hence, f is univalent in D