Answer to Question #125907 in Complex Analysis for Sanjana

Question #125907
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
1
Expert's answer
2020-07-13T18:57:34-0400

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


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