# Tagged Questions

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### Verify that $\sqrt{2}\left\| z \right\| \ge \left|\Re(z)\right| + \left|\Im(z)\right|$

Verify that $\sqrt{2}\left\| z \right\| \ge \left|\Re(z)\right| + \left|\Im(z)\right|.$ I started off noting that $z=x+iy$ and that $Re(z)=x$ and $Im(z)=y$ Then I know that I have to square both ...
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### Triangle Inequality on complex numbers

Problem Let $z= x + iy$, then prove that: $$|x| + |y| \le 2 ^{1/2} |z|$$ Progress I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at ...
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### 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) ...
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### 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.
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### 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 ...
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### 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 ...
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### Inequality in complex numbers

Prove that for all $z\in \mathbb{C}$ $$\frac{\Vert z+i\Vert z\Vert \Vert}{\Vert z+1\Vert}\leq \frac{2\Vert z \Vert}{\Vert z \Vert +1}$$
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### 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 $z_2=z_1+(-z_1)+z_2$. $${}$$
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### 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 ...
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### $|g(t_{1}) e^{-(t_{1}-x)^{2}}- g(t_{2})e^{-(t_{2}-x)^{2}}|\leq |f(t_{1}) e^{-(t_{1}-x)^{2}}- f(t_{2})e^{-(t_{2}-x)^{2}}|$?

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 ...
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### 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 ...
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### 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 $\displaystyle\left|\frac{a+b}{1+a\bar{b}}\right|<1$?
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### 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 ...
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### equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
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### 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 ...
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### 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 ...
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?