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2.9 Pick out the true statements:

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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.

i was trying and thinking this about this problem many times but i could not get it, im thinking that radius of convergence is less than 2, there will beanalytics function less than 2 likely |z| < 1 is assume only real value.

i don't the other option , i have no any other hints solve this question If anybody help me i would be very thankful to him.

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a. is true by the identity theorem.

b. is true by the fundamental theorem of algebra.

c. If $f$ is a non-constant analytic function in the disc $D=\{z: |z| < 1 \}$, then $f(D)$ is open in $\mathbb C$. Hence c. is not true.

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  • $\begingroup$ thanks but how f(D) is open in C i didn't get this line@ fred $\endgroup$ – lomber Aug 14 '17 at 13:00

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