0
votes
1answer
34 views

Complex numbers

If someone could help me with this question I would really appreciate it.For some reason I am getting a weaker version of these inequalities when applying triangle inequality. Let S be the interior ...
2
votes
1answer
39 views

Show inequality of complex number: $|\frac{a+b}{1+a\bar{b}}|<1$

Suppose $a,b\in\mathbb{C},|a|<1,|b|<1$, how to see $\displaystyle\left|\frac{a+b}{1+a\bar{b}}\right|<1$?
0
votes
2answers
31 views

Cauchy-Schwarz in complex case, using discriminant

There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz ...
3
votes
1answer
82 views

A complex number inequality

We have real numbers $p,q,r \gt 0$. Then show that, for all complex numbers $z\neq 0$, $$|z-p|+|z-q\omega|+|z-r\omega^{2}|\gt p+q+r$$ Here, $\omega=e^\frac{i2\pi}{3}$ Actually, I came to this ...
13
votes
2answers
155 views

Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle

Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. The maximum principle or Rouche's theorem can be used to show that there exists a point $b$ on the unit circle such ...
5
votes
1answer
56 views

Simple-looking bound on root of unity

I am trying to prove some bound and stuck with the following: If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$ ($n,N$ are integers) How can I prove it?
1
vote
1answer
37 views

How to show $|(1-z)e^z| \geq e^{-|z|^2}$ for all $|z| \leq 1/2$.

I'm trying to prove the above inequality, but keep running into difficulties. It seems breaking up the RHS into $e^{-x^2}e^{-y^2}$ doesn't help much, and similarly writing LHS $\geq |e^{z}| - ...
4
votes
2answers
79 views

Given $f(z)=z^2+c$. Prove that $|z|>|c|+1$ implies $|f(z)|>|z|$

Consider the quadratic function $f(z)=z^2+c$. If $|z|>|c|+1$, show that $|f(z)|>|z|$. Edit: This is not a homework problem. I found this in my textbook.
1
vote
1answer
84 views

How to prove the following inequality of logarithm?

Let $x,y,z\in\mathbb{C}.$ Suppose $$z=\frac{1}{2}(xy\pm\sqrt{x^2y^2-4(x^2+y^2)} ).$$ Show that $$log^+|z|\leq log^+|x|+log^+|y|+log 2.$$ Where $log^+\phi=max\{0,log\phi\}.$ Here we are also ...
1
vote
1answer
33 views

How find this inequality find the maximum $z_{5}$

let $z_{1},z_{2},z_{3},z_{4},z_{5}\in C$,such $$\begin{cases} |z_{1}|\le 1,|z_{2}|\le 1\\ |2z_{3}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ |2z_{4}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ ...
1
vote
1answer
59 views

How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $

If $z$ is any non-zero complex number, how to prove the following inequality? $$ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $$ Hints please!
0
votes
0answers
133 views

What is the necessary and sufficient condition for the equality in the generalized triangle inequality for complex numbers?

Let $z_1$, ... , $z_n$ be complex numbers, What is the necessary condition for the equality $$ |z_1 + ... + z_n | = |z_1| + ... + |z_n| $$ to hold?
0
votes
1answer
49 views

A complex number inequality and the limitation

How to prove $$|z-1|^\frac{1}{n}\ge||z|^\frac{1}{n} e^{\frac{1}{n}i\arg{z}}-1|,$$ for $n\in \mathbb{N}^+$ and $$|z-1|^\frac{1}{n}-||z|^\frac{1}{n} e^{\frac{1}{n}i\arg{z}}-1|\rightarrow 1$$ as ...
2
votes
1answer
103 views

An inequality about complex numbers

How the prove the following inequality $|z-1|^r\ge|z^r-1|$ holds for some branch of $z^r$. where $0\le r<1$, and $z\in \mathbb{C}$ is a complex number. If such a branch exists will be fine. ...
1
vote
2answers
587 views

Triangle Inequality with complex numbers: Prove that ||x|−|y||≤|x|-|y|.

Prove that $ ||x| - |y|| \le |x| - |y| $ for all $ x,y \in \mathbb{C} $. I fully understand the other inequality: $|x+y| \le |x|+|y| $ for all $ x,y \in \mathbb{C} $. But I have no clue how to start ...
1
vote
1answer
65 views

an inequality about complex number

How to prove that $|a-b|^\gamma\ge||a|^\gamma-|b|^\gamma|$ where $0\le\gamma<1$ and $a,b$ are complex numbers. Is it a famous inequality?
1
vote
1answer
61 views

Complex number inequality, $|e^{z_1}-e^{z_2}| \leq |z_1 - z_2|$ if $Re(z_1),Re(z_2) \leq 0$

I'm trying to show the complex inequality $|e^{z_1}-e^{z_2}| \leq |z_1 - z_2|$ holds if $Re(z_1),Re(z_2) \leq 0$. It seems intuitively obvious but I haven't been able to find something that works. ...
0
votes
1answer
97 views

Inequality related to modulus of complex numbers

How to show: for any $\alpha >0$, there is constant depends on $\alpha$, say $C=C(\alpha)$, such that, $$\mid \mid w \mid ^{\alpha} w - \mid z \mid ^{\alpha} z \mid \leq C (\mid w \mid ^{\alpha} ...
1
vote
1answer
50 views

Complex number inequality

For any $z_1, z_2 \in \mathbb{C}$, is there exist $C>0$ such that $$ 4|z_1|^2 |z_2|^2 + |z_1^2 - z_2^2|^2 \ge C (|z_1|^2 + |z_2|^2)^2 \;\;?$$
2
votes
1answer
50 views

A question of complex numbers inequality $ \overline{z_1}^2 z_2 + z_1^2 \overline{z_2} \leq C ( |z_1|^3 + |z_2|^3) $

Can I find a positive constant $C$ such that $$ \overline{z_1}^2 z_2 + z_1^2 \overline{z_2} \leq C ( |z_1|^3 + |z_2|^3) $$for any complex numbers $z_1, z_2$? Here the overline denotes its complex ...
2
votes
1answer
76 views

Inequality holding for complex numbers in the unit disk

In Nehari's book Conformal Mapping he gives it as an exercise to prove that for $a,b\in \mathbb{C}$, $|a|, |b| <1$ we have $$\frac{|a|-|b|}{1-|ab|} \leq \left|\frac{a-b}{1-\overline{a}b}\right| ...
0
votes
2answers
55 views

Complex number question

For any complex numbers $z_1, z_2$, is the quantity $S$: $$ S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 ...
2
votes
3answers
219 views

Proving an inequality: $|1-e^{i\theta}|\le|\theta|$

We have been using this result without proof in my class, but I don't know how to prove it. Could someone point me in the right direction? $$|1-e^{i\theta}|\le|\theta|$$ I believe this is true for ...
2
votes
2answers
42 views

Small inequality on unit open disc

For $|u|,|z|<1$, $u,z$ complex numbers, how to show the inequality: $|\frac{u-z}{1-\bar uz}|<1$?
0
votes
3answers
62 views

Inequality about sum of complex numbers.

Let ${\alpha_1,\alpha_2,...,\alpha_n}$ be complex numbers, prove that $$|\alpha_1+\alpha_2+\cdots+\alpha_n|^2 \leq n(|\alpha_1|^2+|\alpha_2|^2+\cdots+|\alpha_n|^2).$$
1
vote
1answer
63 views

Lower bound for polynomial with complex coefficient

Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$. ...
0
votes
2answers
207 views

Triangle inequality complex analysis

Using Triangle Inequality, prove if $|z−c| \le |c|/2$, then $|z| \ge |c|/2$.
6
votes
1answer
110 views

An Inequality question

I have the following question. I have to find a $\delta>0$ such that for all complex numbers $x,y$ the following holds true - \begin{equation} \frac{1}{2\pi}\int_0^{2\pi}|x+e^{it}y|\,dt \ge ...
0
votes
3answers
104 views

A not too simple complex number inequality

Prove the following inequality $\forall n>0$ $\forall z \in \mathbb{C}$ such that $|z|=1$: $$\vert z+\frac{1}{z} \vert <\vert z^{n} + i \vert + \vert \overline{z}^{n} + i \vert \leq 2\sqrt{2} ...
2
votes
1answer
85 views

Prove that $x-\mid y\mid\le\left|\frac{x-y}{1-xy}\right|<1$

Let a be a real number, b is a complex number, $a \in (0,1)$ and $|b|<1$ Prove that $$x-\mid y\mid\le\left|\frac{x-y}{1-xy}\right|<1$$ I have solved the left side: ...
1
vote
1answer
108 views

Inequalities with $\sin(z)$, $z \in \mathbb{C}$

An exercise asks to find all $z\in\mathbb C$ such that $|{\sin z}|\leq 1$ and then an $n\in\mathbb N$ such that $|\sin(in)|>10000$. Here are some results we can use. For all $z=x+iy\in\mathbb C$ ...
2
votes
2answers
129 views

upper and lower bounds of a complex expression

How do I prove that $$\sqrt{\frac{7}{2}}\leq |1+z|+|1-z+z^2|\leq 3\sqrt{\frac{7}{6}}$$ for all complex numbers $|z|=1$? I don't really know how to grapple with it. P.. I am extremely sorry, the ...
5
votes
2answers
415 views

Proof of an inequality about $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$

I've encountered an inequality pertaining to the following expression: $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$, where $z$ is a complex number. After writing $z$ as $x + iy$ we have ...
1
vote
2answers
100 views

Proving an inequality concerning arbitary complex numbers

$\def\abs#1{\left|#1\right|}$If $y$ and $z$ are any complex numbers then prove that \[ 2 \abs{y+z}\ge \bigl(\abs y + \abs z\bigr) \abs{\frac y{\abs y} + \frac z{\abs z}} \]
0
votes
3answers
156 views

Complex inequality

How can I show this inequality $\sqrt{2}|z|\geq |\mathrm{Re} (z)|+|\mathrm{Im}(z)| $ please give me some hint. Which result is useful to show this. please help me out.thanks in advance.
5
votes
1answer
163 views

Show $|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$

How to prove for any complex numbers $a$, $b$, $c$, the inequality $$|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$$ is correct?
1
vote
2answers
166 views

An inequality for two complex numbers

I recently saw the following inequality for complex numbers: If $a,b\in\mathbb C$ and $|a + b|$ and $|a-b|$ are each less than or equal to 1, then $$|a| + |b^2|/2 \leq 1.$$ How can one prove this?
3
votes
1answer
118 views

inequality with modulus of complex number

Let $ \displaystyle{ z_1, z_2 \in \mathbb{C} }$ where $ z_1, z_2 \neq 0$ Prove that: $\displaystyle |z_1 +z_2| \geq \frac{1}{2} \left( |z_1|+|z_2| \right) \left|\frac{z_1}{|z_1|} + ...
4
votes
1answer
1k views

Proving the Schwarz Inequality for Complex Numbers using Induction

I want to prove the following version of the Schwarz Inequality for complex numbers $a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $b_1, b_2, \ldots, b_n \in \mathbb{C}$: $$|\sum_{j=1}^n a_j ...
2
votes
2answers
129 views

That $|a|\leq|b|$ implies existence of complex $z$ satisfying $|z-a|+|z+a|=2|b|$?

I'm looking at the equation $|z-a|+|z+a|=2|b|$. If there are complex values $z$ satisfying this equation, then $$ 2|b|=|z-a|+|z+a|=|a-z|+|z+a|\geq|(a-z)+(z+a)|=|2a|=2|a| $$ so $|a|\leq |b|$. ...
1
vote
1answer
538 views

Inductive proof of Cauchy's inequality for complex numbers?

I'm trying to put together an inductive proof of Cauchy's inequality for the complex case, $$ \left|\sum_{i=1}^na_ib_i\right|^2\leq\sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2. $$ The base case is easy, ...
1
vote
1answer
237 views

Inequality for modulus

Let $a$ and $b$ be complex numbers with modulus $< 1$. How can I prove that $\left | \frac{a-b}{1-\bar{a}b} \right |<1$ ? Thank you
4
votes
1answer
161 views

Complex number inequality?

Suppose $|z|>1$ for $z$ a complex number. I'm trying to build a certain comparison test to test convergence. I'm wondering, is it true that $$ \frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1}? $$
13
votes
2answers
568 views

Inequality with Complex Numbers

Consider the following problem: Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge ...
1
vote
4answers
157 views

How to analyze the modulus of $\lambda = (1-2\mu)+2\mu\cos\theta+i\nu\sin \theta$?

Consider the complex number $$ \lambda = (1-2\mu)+2\mu\cos\theta+i\nu\sin \theta $$ where $i=\sqrt{-1}$, $\mu,\nu$ are constants, and $\mu>0$. Question: How can I get that $|\lambda|\leq ...
0
votes
2answers
75 views

Proving a multivariable inequality

I would like to know if it is possible to select R1, R2, R3, C1, and C2 such that the quadratic equation yields complex roots. $s_{1,2}=-\frac{b\pm \sqrt{b^2-4ac}}{2a}$ where $a = ...
4
votes
2answers
108 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
3
votes
3answers
93 views

non-calculus proof that $|x| + |y| \le \sqrt{2} |z|$

I would like to see a nice, non-calculus proof that $|x| + |y| \le \sqrt{2} |z|$, where $z$ is the complex number $x + iy$. The more elementary the better, avoiding even trig if possible. Thank you.
4
votes
1answer
154 views

If $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $|z| \leq (4 \sin(1/4))^{-1} |\sin z|$

I came across the following assertion and am having trouble justifying it: If $z$ is a nonzero complex number with $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $$ \left| \frac{z}{\sin z} \right| ...
7
votes
2answers
265 views

How to show a complex number inequality

A classmate consulted me this problem, after a few moment's thought I found it was difficult, so I wish to try my luck here. Let $z_1,z_2,z_3,z_4\in \mathbb{C}$ such that ...