Archive

Posts Tagged ‘complex’

IMC 2013 Problem 6

Problem 6. Let ${z}$ be a complex number with ${|z+1|>2}$. Prove that ${|z^3+1|>1}$.

Solution: Write ${z}$ in its trigonometric form:

$\displaystyle z=r(\cos \theta + i \sin \theta).$

Then the condition ${|z+1|>2}$ translates to

$\displaystyle r^2+2r\cos \theta +1 >4$

which is equivalent to

$\displaystyle r^2+2r\cos \theta -3 >0.$

Since ${r}$ must be positive, we have as a consequence

$\displaystyle r > \sqrt{\cos^2 \theta+3}-\cos \theta.$

The condition ${|z^3+1|>1}$ is equivalent to

$\displaystyle r^6+2r^3 \cos (3\theta) >0$

which translates to

$\displaystyle r^3+2 \cos (3\theta)>0.$

Thus it is enough to prove that

$\displaystyle (\sqrt{\cos^2 \theta+3}-\cos \theta)^3 \geq -2 \cos (3\theta)=-2\cos \theta (4 \cos^2 \theta -3).$

Note that the result we need to prove is not trivial if and only if ${\theta \in (\pi/6,\pi/2)}$ and on that interval ${\cos \theta}$ is positive.

Denote ${t=\cos \theta}$ then ${t \in (0,\sqrt{3}/2)}$ and we want to prove that

$\displaystyle (\sqrt{t^2+3}-t)^3 +2t(4t^2-3)\geq 0.$

If we denote ${f(t)=(\sqrt{t^2+3}-t)^3 +2t(4t^2-3)}$ then ${f(\sqrt{3/8})=0=f'(\sqrt{3/8})=0}$.

If we evaluate then we obtain

$\displaystyle f(t)=(a^2+3)^{3/2}+3a^2\sqrt{a^2+3}+4a^3-15a$

which is a sum of two convex functions (first two and last two terms) among which one is strictly convex, and therefore is strictly convex (on ${[0,1]}$).

This means that ${\sqrt{3/8}}$ is the global minimum of ${f}$ on ${[0,1]}$ and therefore ${f}$ is positive on that interval which finishes the proof.

Categories: Complex analysis, Olympiad Tags: ,

IMC 2013 Problem 5

Problem 5. Does there exist a sequence ${(a_n)}$ of complex numbers such that for every positive integer ${p}$ we have that ${\sum_{n=1}^\infty a_n^p}$ converges if and only if ${p}$ is not a prime?

Geometric mean for n complex numbers

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$  there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

Romanian TST 2004

Every expansion has a 0

Suppose $f$ is entire such that for each $x_0 \in \mathbb{C}$ at least one of the coefficients of the expansion $f=\sum\limits_{n=0}^\infty c_n (z-z_0)^n$ is equal to 0. Prove that $f$ is a polynomial.

Power series & number theory

Let $F(z)=\sum\limits_{n=1}^\infty d(n) z^n$ where $d(n)$ denotes the number of divisors of $n$. Calculate the radius of convergence of this series and prove that $F(z)=\sum\limits_{n=1}^\infty \frac{z^n}{1-z^n}$. Furthermore, prove that $F(r)\geq \frac{1}{1-r}\log \frac{1}{1-r}$ for $r \in (0,1)$.

Amazing property of entire functions

Prove that if $p,q,r$ are non-constant, non-vanishing entire functions with $p+q+r=0$ then there exists an entire function $h$ such that $p,q,r$ are constant multiples of $h$.

Blaschke factors

Take $w,z \in \mathbb{C}$ such that $\bar{z}w \neq 1$. Prove that $\left| \frac{w-z}{1-\bar{w}z} \right|<1$ if $|z|<1$ and $|w|<1$ and also $\left| \frac{w-z}{1-\bar{w}z} \right|=1$ if $|z|=1$ or $|w|=1$.
Prove that for any $w$ fixed in the unit disk $\mathbb{D}$, the mapping $F_w(z)=\frac{w-z}{1-\bar{w}z}$ has the following properties:
1) $F_w$ maps $\mathbb{D}$ to $\mathbb{D}$ and is holomorphic.
2) $F_w$ interchanges $0$ and $w$.
3) $|z|=1 \Rightarrow |F_w(z)|=1$.
4) $F_w$ is an involution on $\mathbb{D}$ and thus is bijective.
5) Construct a bijective, holomorphic function, which maps the unit disk to itself, takes the unit circle into itself and interchanges two given complex numbers $z,w$ with $|z|<1,|w|<1$.