1
vote
1answer
41 views

show an integral is bounded by a constant independent of a parameter

This is a question in Treves. Suppose $a>1$ and $\tau \in \mathbb R $, (i) show that for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge(\tau ^2+|\xi|^2+a^2)^{1/2}$ (ii) ...
2
votes
1answer
30 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
1
vote
1answer
53 views

A simple complex inequality

I feel this is not hard, but no way to prove it $|\sqrt{z^2 -4}-z|\le 2$ Any body can help? Thanks! The total statement should be one of the branchs of square root should satisfy this ...
1
vote
1answer
35 views

How find the minimum of the $|w^3+z^3|$,if $|z+w|=1,|z^2+w^2|=14$

let complex $z,w$ such $$|z+w|=1,|z^2+w^2|=14$$ find the minimum of the value $$|w^3+z^3|$$ My idea: let $$z=a+bi,w=c+di\Longrightarrow z+w=(a+c)+(b+d)i,z^2+w^2=(a^2+b^2+c^2+d^2)+2(ab+cd)i$$ then we ...
3
votes
2answers
55 views

Inequality in complex numbers

Prove that for all $z\in \mathbb{C}$ $$\frac{\Vert z+i\Vert z\Vert \Vert}{\Vert z+1\Vert}\leq \frac{2\Vert z \Vert}{\Vert z \Vert +1}$$
2
votes
3answers
47 views

How can one prove $||z_1|-|z_2||\le|z_1+z_2|$?

How can it be shown that for the complex numbers $z_1$ and $z_2$: $$||z_1|-|z_2||\le|z_1+z_2|$$ My text provides a hint that $z_1=z_1+z_2+(-z_2)$, and $z_2=z_1+(-z_1)+z_2$. $${}$$
0
votes
1answer
24 views

An inequality on the real part of a square root

I have the following inequality: $\Re(k+z) \geq \Re \sqrt{(k+z)^2-4z}$ where $k$ is real and $z$ complex. Under what conditions on $k$ and $z$ is this inequality true? I suspect that it is true for ...
2
votes
0answers
20 views

Cyclic sum inequality involving five numbers with modulus one and zero sum

When working on this MSE question, I was led to conjecture the following : If $z_1,z_2,z_3,z_4,z_5$ are five complex numbers with modulus $1$, such that $z_1+z_2+z_3+z_4+z_5=0$, then $$ ...
0
votes
2answers
69 views

Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
0
votes
2answers
17 views

$\big| |z|-|w|\big| \leq |z-w| \implies \big|c_{1} |z|- |w|c_{2}\big| \leq |c_{1}z- c_{2} w| ? $ ($z, w \in \mathbb C, C_{1}, C_{2} >0$)

By triangle inequality, we get, $$\big| |z|-|w|\big| \leq |z-w|; (z, w\in \mathbb C.)$$ Take any $C_{1}, C_{2} > 0$ and fix it. My Question is: Can we expect: $$\big|C_{1} |z|- |w|C_{2}\big| ...
1
vote
1answer
19 views

$|g(t_{1}) e^{-(t_{1}-x)^{2}}- g(t_{2})e^{-(t_{2}-x)^{2}}|\leq |f(t_{1}) e^{-(t_{1}-x)^{2}}- f(t_{2})e^{-(t_{2}-x)^{2}}| $?

Suppose $f, g: \mathbb R \to \mathbb C$ such that $|g(t_{1}) -g(t_{2})| \leq |f(t_{1})- f(t_{2})| $ for every $t_{1}, t_{2} \in \mathbb R.$ Take any $x\in \mathbb R$ and fix it. Edit: We also assume ...
0
votes
1answer
45 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
41 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
38 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
91 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
162 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
57 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
40 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
80 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
34 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
64 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
156 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
53 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
109 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
627 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
68 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
64 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
111 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
55 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
53 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
83 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
57 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
237 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
47 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
71 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
66 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
221 views

Triangle inequality complex analysis

Using Triangle Inequality, prove if $|z−c| \le |c|/2$, then $|z| \ge |c|/2$.
6
votes
1answer
114 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
111 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
110 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
142 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 ...
1
vote
2answers
276 views

Cauchy-Schwarz Inequality for Complex Numbers

Simple question: do we really need the conjugate in the inequality? $$ |\sum_{j=1}^n a_j \overline{b_j}|^2 \leq \sum_{j=1}^n |a_j|^2 \sum_{j=1}^n |b_j|^2 $$
5
votes
2answers
419 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
101 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
163 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
164 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?
3
votes
4answers
2k views

Does the inequality $|\sin(x)|\leq |x|$ extend to the complex numbers?

As it is well known: $$|\sin(x)|\leq |x| \forall x \in \mathbb{R}.$$ Now, if we have a complex number $z$; can I preserve the same inequality $$|\sin(z)|\leq |z|\quad \forall z \in \mathbb{C}?$$
1
vote
2answers
175 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?