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I need to tell whether

  1. $\exists f$ non constant,entire, with $f(0)=e^{i\alpha},|f(z)|={1\over2}\forall z\in\partial\mathbb{D}$

False due to Maximum Modulas Principle

  1. $\exists f$ non constant,entire, with $f(e^{i\alpha})=3,|f(z)|=1\forall |z|=3$

False due to Maximum Modulas Principle

  1. $\exists f$ non constant,entire, with $f(0)=1, f(i)=0,|f(z)|\le 10\forall z\in\mathbb{C}$

False, due to Liouvilles

  1. $\exists f$ non constant,entire, with $f(0)=4-3i,|f(z)|\le 5\forall z\in\mathbb{D}$

False, due to Liouvilles

  1. $\exists f$ non constant,entire, with $f(z)=0\forall z=n\pi$

True, $f(z)=\sin z$

am i right in every case?

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  • $\begingroup$ What is your question? You make a bunch of claims (some of which are a little difficult to read due to excessive and somewhat non-standard use of quantifiers), but I see no question in the text. $\endgroup$ – mrf Dec 8 '14 at 15:21
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  • Item 1. and 2. are correct provided that $\alpha$ is a real number.
  • 3. is correct.
  • 4. Careful: the assumption of boundedness is only on the unit disk and not the whole complex plane. Since the modulus of $f(0)$ is $5$, we can however conclude by the maximum principle.
  • 5. It is correct.
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