# Tagged Questions

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### What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
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### Soft Question: Inequalities like this

I am studying signed and complex measure and at a point in a proof the following lemma is being used: Lemma. If $z_1,...,z_n$ are complex numbers, then there exists a subset $S\subset\{1,2,...,n\}$ ...
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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 ...
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### complex sum-integration

Let $a_n \in \mathbb{R}$. Show that, $$0\le \sum_0^N\sum_0^N\frac{a_na_m}{n+m+\frac{1}{2}}\le \pi \sum_0^Na_n^2.$$ Hint: Integrate on the semicircle with radius R, the sum ...
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### What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where $$\frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v))$$ where I am interested in the ...
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### Find the complex $z$ such $\max{(|1+z|,|1+z^2|)}$ is minimum

find the complex $z$,such $$\max{(|1+z|,|1+z^2|)}$$ is minimum My try: let $z=a+bi$,then $$|1+z|=\sqrt{(a+1)^2+b^2}$$ $$|1+z^2|=|1+a^2+2abi-b^2|=\sqrt{(1+a^2-b^2)^2+4a^2b^2}$$ Then I can't,Thank ...
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### How to rigorously show that $\frac{z^4}{1+z^6}$ behaves like $\frac{1}{z^2}$ for large $|z|$?

I want to make the integral $$\int_{C_R}\frac{z^4}{1+z^6}dz$$ disappear, but I just realised I'm having trouble writing down a coherent set of inequalities that would allow me to bound it by ...
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### An inequality involving arctan of complex argument

I have the following conjecture: $$\text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0.$$ Which seems to be ...
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Let $D := \{z \in \mathbb{C}: |z| < 1\}$ and $f\colon D \rightarrow \mathbb{C}$ be holomorphic. Suppose $\lvert f(z)\rvert \leq 1$ on $D$, show that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| ... 2answers 74 views ### For analytic f on D_2(0) with |f(z)| \le |\sin z| on \partial D_2(0) , show |f(\frac{\pi}{2})| \le \frac{4}{\pi} Let f be analytic on D_2(0) and continuous up to the boundary with |f(z)| \le |\sin z| on \partial D_2(0). Prove that |f(\frac{\pi}{2})| \le \frac{4}{\pi}. This problem appears on an old ... 1answer 81 views ### Inequality holding for complex numbers in the unit disk In Nehari's book Conformal Mapping he gives it as an exercise to prove that for a,b\in \mathbb{C}, |a|, |b| <1 we have$$\frac{|a|-|b|}{1-|ab|} \leq \left|\frac{a-b}{1-\overline{a}b}\right| ...
Let $C$ be a circle of radius $1$ in the complex plane with $n$ points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other $n$ points. ...