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Denote $D=\{z\in\mathbb C:|z|<1\}.$

1) Let $a\in\mathbb C$ with $|a|<1$ and $p(z)=\dfrac a2+(1-|a|^2)z-\dfrac{\overline a}2z^2.$ Show that for all $z\in D$ is $|p(z)|\le1.$

2) Characterize all the $f$ entire functions so that for each $z\in\overline D^c$ satisfy $\left| {f(z)} \right| \le {\left| z \right|^5} + \dfrac{1}{{{{\left| z \right|}^5}}} + \dfrac{1}{{{{\left| {z - 1} \right|}^3}}}.$

3) Let $w_1,w_2\in\mathbb C$ two $\mathbb R-$linearly independent numbers. Show that if $f\in\mathcal H(\mathbb C)$ is so that for each $z\in\mathbb C$ and $f(z+w_1)=f(z)=f(z+w_2),$ then $f$ is constant.

4) Let $\mathcal U\subset\mathbb C$ open and $z_0\in\mathcal U.$ Suppose that $f$ is continuous on $\mathcal U$ and analytic on $\mathcal U-\{z_0\}.$ Show that $f$ is analytic on $\mathcal U.$

1) I think I need to use the Maximum Modulus Principle, but I don't see how.

2) If I let $|z|=R$ then $\left| {f(z)} \right| \le {\left| R \right|^5} + \dfrac{1}{{{{\left| R \right|}^5}}} + \dfrac{1}{{{{\left| {R - 1} \right|}^3}}},$ but $f$ was given entire so it has convergent Taylor series and by using Cauchy's integral formula I can conclude that $f^{(k)}(0)=0$ for some $k\ge n,$ and then functions $f$ are polynomials of degree $n-1,$ does this make sense?

3) I think I could use Liouville here, but I don't have that $f$ is entire, but $f$ is periodic, right?, and a periodic entire function is bounded so I could conclude by using Liouville, but I don't have that $f$ is entire. Perhaps there's another way on doing this.

4) I think I should use a remarkable theorem here but I don't remember, it looks hard.

1) Let $a\in\mathbb C$ with $|a|<1$ and $p(z)=\dfrac a2+(1-|a|^2)z-\dfrac{\overline a}2z^2.$ Show that for all $z\in D$ is $|p(z)|\le1.$

2) Characterize all the $f$ entire functions so that for each $z\in\overline D^c$ satisfy $\left| {f(z)} \right| \le {\left| z \right|^5} + \dfrac{1}{{{{\left| z \right|}^5}}} + \dfrac{1}{{{{\left| {z - 1} \right|}^3}}}.$

3) Let $w_1,w_2\in\mathbb C$ two $\mathbb R-$linearly independent numbers. Show that if $f\in\mathcal H(\mathbb C)$ is so that for each $z\in\mathbb C$ and $f(z+w_1)=f(z)=f(z+w_2),$ then $f$ is constant.

4) Let $\mathcal U\subset\mathbb C$ open and $z_0\in\mathcal U.$ Suppose that $f$ is continuous on $\mathcal U$ and analytic on $\mathcal U-\{z_0\}.$ Show that $f$ is analytic on $\mathcal U.$

__Attempts__:1) I think I need to use the Maximum Modulus Principle, but I don't see how.

2) If I let $|z|=R$ then $\left| {f(z)} \right| \le {\left| R \right|^5} + \dfrac{1}{{{{\left| R \right|}^5}}} + \dfrac{1}{{{{\left| {R - 1} \right|}^3}}},$ but $f$ was given entire so it has convergent Taylor series and by using Cauchy's integral formula I can conclude that $f^{(k)}(0)=0$ for some $k\ge n,$ and then functions $f$ are polynomials of degree $n-1,$ does this make sense?

3) I think I could use Liouville here, but I don't have that $f$ is entire, but $f$ is periodic, right?, and a periodic entire function is bounded so I could conclude by using Liouville, but I don't have that $f$ is entire. Perhaps there's another way on doing this.

4) I think I should use a remarkable theorem here but I don't remember, it looks hard.

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