All Questions
Tagged with trigonometric-sums inequalities
13 questions
21
votes
3
answers
2k
views
Trigonometric inequality
For odd and coprime positive integers $p$ and $q$, the following inequality holds:
$$\sum_{m=1}^{p} \sum_{n=1}^{q} \frac{2}{\cos(\frac{2m\pi}{p})+\cos(\frac{2n\pi}{q})} \le pq(|p-q|+1)$$
Unfortunately,...
- 683
21
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
5
votes
1
answer
244
views
Smallest regular $m$-gon covering a regular $n$-gon
I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.
Let us fix a regular $n$-gon with area $1$. What is the smallest ...
- 1,889
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
18
votes
3
answers
1k
views
Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$
Let $x_{1},x_{2},\cdots,x_{n}>0$, show that
$$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2+\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le \left(2+\dfrac{n}{4}\right)\sum_{k=1}^{n}x^2_{k}$$
This ...
- 3,153
2
votes
0
answers
511
views
Conjecture about $\prod_{n=1}^{\infty}\left|1+\frac{\tan\left(n\right)+1}{n\tan\left(n+1\right)+n}\right|=0$
Ps: I have reached the number of question I can propose so let this here please .And it's a work on open questions .
Does we have in radians :
$$\prod_{n=1}^{\infty}\left|1+\frac{\tan\left(n\right)+1}{...
0
votes
0
answers
125
views
$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \sin(\mu E)\cos(\mu F) \geq 0$ - Version 2.0
I recently asked this question here Inequality involving sine and cosine
It turned out that with the conditions I required there, the inequality does not hold. I tried to add extra conditions now, ...
- 447
3
votes
1
answer
771
views
Inequality involving sine and cosine
I am trying to prove that given $A,B,C,D,E,F \in \big]0,\frac{\pi}{2}\big]$ fixed and $A+C \geq E$ and the following equation holds for $\mu = 1$:
$$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \...
- 447
1
vote
1
answer
138
views
Proof of an inequality $s_m(n) \le f_m(n)$
For fixed $m = 0, 1, 2, ...$
$$f_m(k) = \prod_{j=1}^{m}(k+j).$$
Some examples of $f_m(k)$ are as following:
$$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$
The $s_m(n)$ is defined as ...
- 161
3
votes
2
answers
231
views
Inductive proof of $s(n)≤n+1$
I was able to conclude, numerically, the following:
$$s(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{(k+1)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2}\le n+1$$
for $x\in[0,1]$. For example
...
- 161
5
votes
3
answers
787
views
positive sum of sines
This was asked but never answered at MSE.
Let $f(x) = \sin(a_1x) + \sin(a_2x) + \cdots + \sin(a_nx)$, where the $a_i$'s
represent distinct positive integers. Suppose also that $f(x)$ satisfies the ...
- 1,411
2
votes
0
answers
182
views
Why hexagons? The maximal minimum of a sum of cosines on the plane with frequencies on the unit circle
We wish to maximize the minimum of a weighted sum of cosines in the plane, when the frequency components are on the unit circle. Formally:
$$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) \...
- 968
2
votes
1
answer
222
views
trigonometric sum and inequalities
let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\...
- 45