All Questions
Tagged with cv.complex-variables inequalities
45 questions
1
vote
1
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63
views
Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))
To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients
$...
- 111
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
3
votes
2
answers
645
views
Upper bound for complex integral
I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...
- 43
5
votes
1
answer
229
views
An inequality for polynomials
I have been thinking about the validity of the following inequality: if $P(z)=\sum_{k=0}^na_kz^k, a_n\neq 0$ and $P(z)$ is non-zero in $|z|<1, $ then for $\theta \in [0, 2\pi],$ and $p>0$
\...
- 559
6
votes
1
answer
468
views
a problem in complex-variable inequality
Let $n\ge2$ be a given positive integer, and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such
$$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n.$$
Prove or disprove
$$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq ...
- 4,280
4
votes
1
answer
183
views
Maximal increasing behaviour of $\sqrt{-\log x}\operatorname{Li}_{1/2}(x)$
This is an extension of a problem in mathematical biology. It appears that
For any $\varepsilon>0$, there exists an interval $I\subset(0,1)$ on which $\sqrt{-\log x}^{1+\varepsilon}\operatorname{...
- 1,459
3
votes
1
answer
137
views
Estimate the homogeneous components of a polynomial against its maximum
Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over ...
- 463
46
votes
3
answers
5k
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Putnam 2020 inequality for complex numbers in the unit circle
The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020:
Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
- 33.8k
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$
where $y\in(0,\infty)$, $u\in(0,1)$, and
$$f(t):=t+\pi (1-t) t \cot (\pi t).$$
Here are the graphs of $f$ (black), ...
- 128k
4
votes
1
answer
488
views
Linear combinations of roots of unity
Suppose that $x_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form
$$ z = \sum_{i=0}^{n-1} x_i \omega^...
- 2,222
6
votes
1
answer
182
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,135
12
votes
1
answer
858
views
Is this function concave?
Let
$$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$
for $u>0$. Is the function $h$ concave on $(0,\infty)$?
(For context, see Proposition 4.4.4 and formula (4.4.21) in this ...
- 128k
3
votes
0
answers
320
views
is $\sum_{i=1}^{n}\sum_{j=1}^{n}|a_{i}+a_{j}|\cdot p_{i}p_{j}\ge \sum_{i=1}^{n}p_{i}|a_{i}|$?
see:old post
and Make the integral discrete, we have
conjecture :for any complex numbers $a_{i},i=1,2,\cdots,n$,and $p_{i}\ge 0$ such $p_{1}+p_{2}+\cdots+p_{n}=1$,then maybe be have
$$\sum_{i=1}^{n}\...
- 4,280
11
votes
2
answers
1k
views
A problem in additive combinatorics
$\color{red}{\mathrm{Problem:}}$
$n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
- 302
1
vote
0
answers
199
views
What is decoupling theory means on Tao Blog ? And what is its purpose in mathematics? [closed]
I accrossed on Tao Blog a new theory for me which it is called "Decoupling theory", But I didn't find in the web its definition and its purpose , I find only this article in wiki but this very far ...
user147204
3
votes
1
answer
237
views
Isoperimetric inequality for analytic functions on an annulus
Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
- 434
5
votes
1
answer
404
views
Entire function not less than $\sqrt{\sinh x/x}$ on the real line
If the answer to this question is in affirmative, then this would yield a good answer to this question.
Let $f\colon \mathbb C\to\mathbb C$ be an entire function whose values on the real line are ...
- 23.7k
2
votes
1
answer
214
views
Ratio of hypergeometric function
Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function?
$$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$
for $1>x>y>0$ ideally in the form like some ...
- 1,495
3
votes
1
answer
223
views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
1
vote
0
answers
87
views
An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
- 4,135
17
votes
1
answer
1k
views
A conjecture of Littlewood
The following is a conjecture due to Littlewood.
For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality
$$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$ holds....
- 1,583
4
votes
1
answer
269
views
An inequality of T. Carleman
I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...
- 571
3
votes
0
answers
237
views
best-possible inequalities for hypergeometric functions
In what follows, let $n$ be a positive integer and $0<a<1/2$. I am interested in the Gauss hypergeometric functions, $_{2}F_{1}( -n, -n-a; 1-a; z)$. Notice that these are polynomials, if that is ...
- 113
5
votes
1
answer
162
views
Supremum norm of certain quantity
Is there any easy way of finding supremum of the quantity $$\sum_{i,j=1}^n|z_i-z_j|,$$ where $|z_i|=1$ for $1\leq i\leq n$ ? We are considering complex variables of course.
- 455
10
votes
2
answers
886
views
An attempt to generalize the previous inequality
In my previous MO question, the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps ...
- 43.2k
4
votes
1
answer
1k
views
seeking proofs: infinite series inequalities
Question. Numerically, the following is convincing. However, is there a proof?
$$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4
<\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\...
- 43.2k
3
votes
2
answers
283
views
complex polynomials and inequalities
Let $z_1,z_2,\dots,z_n\in\Bbb{C}$ be distinct and $w_1,w_2,\dots,w_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that
$$f(z_j)=w_j,\qquad g(z_j)=\bar{w}_j \...
- 43.2k
0
votes
1
answer
172
views
An inequality in product space $V$ [closed]
I found an inequality as following: Let $x, y, z$ be three complex numbers then:
\begin{equation*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation*} (1)
The ...
- 1,044
5
votes
1
answer
330
views
maybe this conjecture also hold to this complex inequality
I have solve this following
Question: Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}...
- 4,280
4
votes
1
answer
1k
views
Analytic Combinatorics: upper bound for sum of absolute values of two complex functions: $|z f'(z)| + |2 f(z) - zf'(z)| \leq 2f(|z|)$
Let $f \colon \mathbb C \to \mathbb C$ be a complex-valued analytic function with non-negative coefficients of Taylor series at 0 (suppose that radius of convergence is $+\infty$ for simplicity):
$$
...
- 247
0
votes
2
answers
402
views
Absolute value inequality with complex numbers
Following a problem I found on mathstack, with no solution, and no comment, so I think this inequality is not easy, so I post it here (because I think there are more some good math job, maybe someone ...
- 4,280
4
votes
1
answer
1k
views
upper bound on derivatives of a function defined on an arc
This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.
You can skip examples below and read from "General setting" at ...
- 398
2
votes
0
answers
153
views
system of complex number equations
Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...
- 309
2
votes
0
answers
102
views
Octahedron and System of trigonometric equations
Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
$$\sum_{k=...
- 309
14
votes
2
answers
534
views
regular polygon question
Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...
- 309
1
vote
0
answers
237
views
Asymptotics to Taylor expansions?
I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
https://math.stackexchange.com/questions/1440931/...
- 763
2
votes
0
answers
158
views
$\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?
We have $\frac{1}{2} < \sigma < 1$ and
$$
f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|
$$
. My goal is proving this statement that $|f(n)|$ is $O(\...
- 1,113
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
6
votes
1
answer
536
views
Bound on the sum of arguments
Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...
- 128k
47
votes
3
answers
6k
views
Absolute value inequality for complex numbers
I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...
- 953
4
votes
1
answer
235
views
A homogeneous but slightly asymmetric inequality
I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$
\begin{equation}
\left|\left|\sum_{j=1}^l z_j\right|^p-\sum_{j=1}^l\left|z_j\right|^...
- 403
4
votes
1
answer
287
views
Inequality for certain analytic functions
Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y_0$ we have
$$
\left| {f\left( {x + iy} \right)} \right| \le f\left( x \right)
$$
for any ...
- 232
0
votes
1
answer
106
views
What is the corresponding version in the complex space of this proposition got in the real space real
How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x_1,...,x_n)\in C^n$ ?
suppose that $\...
- 1
2
votes
1
answer
465
views
Complex version of Farkas' lemma
It is well known result of linear optimization theory that, if a finite set $S$ of linear inequalities in real variables $x_1,x_2, \ldots ,x_n$ implies a linear inequality $i$ in $x_1,x_2, \ldots ,...
- 3,595
3
votes
5
answers
11k
views
What is the angle between two complex vectors?
Let $x, y\in R^n$ and $x, y$ are nonzero, it is well known
$\frac{x^Ty}{\parallel x\parallel_2\parallel y\parallel_2}(\parallel x\parallel_2+\parallel y\parallel_2)\le \parallel x+y\parallel_2$. How ...
- 1,858