- Given power series $\sum_{n=0}^{\infty} a_{n} z^{n}, a_{n} \in \mathbb{C}, n=0,1,2, \ldots$
(1) Show that if the power series converges uniformly in the unit disk $D={z \in \mathbb{C}:|z|<1}$, then
$$
\lim \sup {n \rightarrow \infty}\left|a{n}\right|=0
$$
(2) Give an example of a sequence $\left{a_{n}\right}$ with $\lim \sup {n \rightarrow \infty}\left|a{n}\right|=0$ such that the power series converges in the unit disk, but not uniformly. - Consider a continuous function $f:[0,1] \rightarrow \mathbb{C}$, and a function $F: \mathbb{C}-[0,1] \rightarrow \mathbb{C}$ that is defined by
$$
F(z)=\frac{1}{2 \pi i} \int_{0}^{1} \frac{f(t) d t}{t-z}, z \in \mathbb{C}-[0,1]
$$
(1) Prove that $F$ is analytic on $\mathbb{C}-[0,1]$
(2) Prove that
$$
\lim _{h \rightarrow 0+}(F(x+i h)-F(x-i h))=f(x), \forall x \in(0,1)
$$ - (1) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function with the property
$$
|f(z)| \leq c\left(1+|z|^{\frac{7}{2}}\right), z \in \mathbb{C}
$$
for some constant $c$. Prove that $f$ is a cubic polynomial.
(2) Let $D$ be the unit disk in $\mathbb{C}$ and $f: D \rightarrow \mathbb{C}$ be an analytic function with the property that
$$
\begin{array}{r}
f(0)=1, f^{\prime}(0)=3, f^{\prime \prime}(0)=4 \
f^{(n)}(0)=1, \forall n \geq 3
\end{array}
$$
Show that $f$ is unique and obtain its expression. - (1) Show that there exists an analytic function $f$ on the unit disk $D={z \in \mathbb{C}:|z|<1}$ satisfying
$$
f\left(\frac{1}{n}\right)=\frac{1}{n+1}, n=2,3,4, \ldots
$$
(2) Show that there is no analytic function $f$ on the unit disk $D={z \in \mathbb{C}:|z|<1}$ satisfying
$$
f\left(\frac{1}{n}\right)=\frac{1}{n+2}, n+2,3,4, \ldots
$$ - Let $D$ be the unit disk in $\mathbb{C}$ and $f: D \rightarrow \mathbb{C}$ be an analytic function which extends to a continuous function on the closure $\bar{D}$ of $D$.
(1) Show by Cauchy integral formula that $\forall z_{0} \in D, \exists$ a constant $c$ depending only on $z_{0}$ such that
$$
\left|f\left(z_{0}\right)\right| \leq c \sup {|z|=1}|f(z)| $$ (2) By applying the estimate in (1) to $f^{n}$ with $n \geq 1$ an integer, and letting $n \rightarrow \infty$, conclude that we can take $c=1$ in (1). (3) Prove $\exists$ a constant $c^{\prime}$ depending only on $z{0}$ such that
$$
\left|f^{\prime}\left(z_{0}\right)\right| \leq c^{\prime} \sup {|z|=1}|f(z)| $$ (4) Provide an counterexample to show that we cannot in general take $c^{\prime}$ independent of $z{0}$ in (3).
