# Tagged Questions

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### $f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
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### On Cauchy-Riemann equations

Given $f:\mathbb C\to \mathbb C$ is a non-constant entire function. Then which of the following is possible? Re $f(z)=$ Im $f(z)$, Im$\,f(z)<0$, Re$\,f(z)$ is bounded, $f(z)\neq 0,$ for all ...
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### If $f$ is holomorphic and satisfies $f'' = f$ then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic on a disk centered at the origin and $f$ satisfies the differential equation $f'' = f$. Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for constants $A$ and ...
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### let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
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### Show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$

This is a homework problem. For $|z| \le R$ and $|a_j| < R$ for $j=1,\ldots, k$, not all zero, show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$. ...
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### Zeros of analytic function and limit points at boundary

Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose ...
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### Question Relating with Open Mapping Theorem for Analytic Functions

This problem is taken from Section VIII.4 of Theodore Gamelin's Complex Analysis: Let $f(z)$ be an analytic function on the open unit disk $\mathbb{D}=\{|z|<1\}$. Suppose there is an annulus ...
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### Problem based on Schwarz Lemma

Let $D={z\in\mathbb{C}:|z|<1}$ and $f:D\to D$ be analytic with f(0)=0 (i) Show that $|f(z)+f(-z)|\leq2|z|^2$ (ii) Suppose that $|f(z_0)+f(-z_0)=2|z_0|^2$ for some $z_0\in\mathbb{C}\setminus{0}$. ...
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### Riemann Zeta Function Manipulation

The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$ (i) show that for $Re z> 1$, we have (1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
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### Show that an analytic function in the unit disc satisfies a inequality

Question: Let $f$ be an analytic function in the unit disc $D={z\in C: |z|<1}$. Consider a point $z_0\in D$. Show that there must be a positive integer n such that the n-th derivative of $f$ at ...
### Show that if $f$ is analytic in the unit disc then an integer $n$ such that $f(1/n)$ does not equal $1/(n+1)$
This is a variant of question Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$. (i). Show that if $f$ is analytic in the ...
### If $f$ is a non-constant analytic function on a compact domain $D$, then $Re(f)$ and $Im(f)$ assume their max and min on the boundary of $D$.
This is a homework problem I got, my attempted proof is: Since $f$ is non constant and analytic, $f=u(x)+iv(y)$ where neither $u$ nor $v$ is constant(by Cauchy Riemann equations) and $u v$ are both ...