# Tagged Questions

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### When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
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### Why does $2x_1x_2y_1y_2 \leq x_1^2y_2^2+x_2^2y_1^2$?

When I tried to prove the triangle inequality $|z_1+z_2| \leq |z_1| + |z_2|$ algebraically for complex variables $z_1$ and $z_2$, I came across this inequality and found that this is always true no ...
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### 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 ...
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### 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!
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### 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?
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### 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 ...
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### 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. ...
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### 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 ...
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### 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?
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### 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. ...
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### 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: ...
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### 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 $z=x+iy\in\mathbb C$ ...
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### 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 ...
Simple question: do we really need the conjugate in the inequality? $$|\sum_{j=1}^n a_j \overline{b_j}|^2 \leq \sum_{j=1}^n |a_j|^2 \sum_{j=1}^n |b_j|^2$$
### 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 ...