Questions on proving and manipulating inequalities.

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Inequality in inverse Laplacian

I have the following problem, which is motivated by geometric diffusion on a directed graph. Conjecture. Let $A \in [0,1]^{n\times n}$ be strictly substochastic - i.e. $\forall i ~ \sum_j A_{i,j} ...
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A simple elementary inequality (without ABC Thm)

I want to solve this following inequality. $$\sum_{sym } x^5 - 7 \sum _{sym} x^4y + 7 \sum_{sym} x^3 y^2 + 10 \sum_{sym} x^3 yz - 11\sum_{sym} x^2y^2z \ge 0 $$ whenever $ x,y,z \ge 0 $ I do not want ...
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$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
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42 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
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Proving the inequality $3+8\sum_{cyc} {\frac{b^2c^2(3b^2+3c^2+7a^2)}{7a^2+6bc}}\ge 9(a^2+b^2+c^2)$

Prove the following inequality: $$3+8\sum_{cyc} {\frac{b^2c^2(3b^2+3c^2+7a^2)}{7a^2+6bc}}\ge 9(a^2+b^2+c^2)$$ $ ab+bc+ca=3$ and $a,b,c\ge0$ The inequality is really hard so I have not even a ...
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An integral inequality

Suppose $K\in C[0,1]^2$, $G\in C[0,1]$ are arbitrary and given. The question is that does there exists $H\in B[0,1]$ continuous a.e. with possibly finitely many discontinuities such that $$ ...
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51 views

An Inequality $\dfrac{1-x^{2n+1}}{1-x}\geq (2n+1)\ x^{n}$

Show : $${\displaystyle \forall\ n \in \mathbb{N}^* \quad \forall\ x \in [0,1]\cup (1,+\infty)\quad \dfrac{1-x^{2n+1}}{1-x}\geq (2n+1)\ x^{n} }$$ My attempt: Method 1: Case $x=0$, $$1\geq0$$ ...
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62 views

How prove this $\left[\frac{n}{\sqrt{3}}\right] +1\ge \frac{n^2}{\sqrt{3n^2-\frac{50}{9}}}$

For any integer $n\ge 2$ ,Prove that: $\left[\frac{n}{\sqrt{3}}\right] +1\ge \frac{n^2}{\sqrt{3n^2-k}}$ . when $k=\dfrac{50}{9}$ This $k$ is folowing idea found it. let $n=2$,then we have ...
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Solving inequality with multiple inequality signs.

I'm having trouble understanding what method to use when solving the following type of inequality: Find all values for x $$\frac{4x-33}{4x+33} < \frac{x+1}{x-1} \le \frac{25+4x}{25-4x} $$ I ...
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36 views

Inequality involving incomplete gamma function

When trying to answer this question: Find minimum $n$ such that $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$ has no answer inside the circle of radius $100$ centered at the origin I ended up in what ...
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A question related to Integral and supremum

Let $f\in L_{p}([0,1])$ and 1-periodic on $R^{1}.$ Suppose $[a,c]\subset [0,1].$ Are the following quantities equal? $$ \underset{|h|\leq \delta_{1}}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ ...
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Does Hlawka Inequality follow from Triangle Inequality?

On MathOverflow I saw this inequality. Let $E$ is a normed linear space. $$ \|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E $$ Apparently this is always true if $E = ...
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33 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
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63 views

Operator on the space of square summable sequences

We define an operator $T:\mathcal{l}^2(\mathbb{Z})\rightarrow\mathcal{l}^2(\mathbb{Z})$ where $\mathcal{l}^2(\mathbb{Z})$ is the Hilbert space of square summable functions, such that for ...
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67 views

how to show the following two inequalities? (using Euler method)

I'm stuck again on a numerical Problem. In the course numerical solution of ODE we already introduced the Euler method and now we have to show based on this Cauchy Problem: $y'(t)=y(t)^2$ $y(0)=1$ ...
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53 views

Tightness of an inequality

I have an inequality with $a_n>0\forall n$ and $A_n\geq a_n\forall n$ that \begin{equation} \sum^N_{n=1}a_n\frac{a_n}{A_n}\geq \frac{(\sum^N_{n=1}a_n)^2}{\sum^N_{n=1}A_n} \end{equation} however I ...
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“Balancing” Sums

Given are $x_1,\ldots, x_n\in \{0,1,\ldots,n\}$, $y_1,\ldots, y_n\in \{0,1,\ldots,n\}$ with the property that $$\sum_{i=1}^{n}{x_i}\leq B,$$ $$\sum_{i=1}^{n}{y_i}\leq B$$ Let's assume that $B$ is ...
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35 views

How are the Stirling-based bounds for the factorial function proven?

According to (26) on wolfram mathworld, one has $$\sqrt{2\hspace{-0.04 in}\cdot \hspace{-0.04 in}\pi} \cdot n^{n+(1/2)} \cdot \operatorname{exp}((-n)\hspace{-0.02 in}+\hspace{-0.02 ...
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inequality based word problem.

$A$ can complete a piece of work in $16$ days and $B$ can complete in $x$ days. if $A$ and $B$ start working on alternate days, they together complete the work in same number of days irrespective of ...
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122 views

The geometry of a spiral made of adjacent right triangles

In the above figure (not sure if you can see it clearly or not), while using the old standard technique of plotting irrational numbers on number line, I saw this property. If we go on plotting ...
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51 views

Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
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37 views

Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n $ in a textbook,provided that either $e$ or $n$ is large. We ...
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28 views

Generating bounds on $e^x$

I had a question in which I had to find the value of zint of $e^x$. How can I generate bounds on $e^x$ so as to obtain its zint? (zint is floor function which is the greatest integer less than or ...
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29 views

Rotations and inequalities

As a follow up to my question on LQ decomposition and inequalities, I'm trying to explore what effect givens rotations have on a system of inequalities. Suppose I have the simple system $\mathbf{x} ...
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L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
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How to compare the dimensions of two blocks?

Consider I have the dimensions of two boxes (length x width x height), what would be the easiest way to compare them, allowing N% error? For example, 20x30x40 would be the same box as 40x30x20, so ...
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146 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
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Find condition on $A$ and $B$ so that $|q_o|+|q_1|<1$.

I have to find condition on $A$ and $B$ so that $$|q_o|+|q_1|<1$$ Where $$q_0=\frac{-1-4A+C+(2B+1)\sqrt{8A+(B+C-1)^2}-B(2B+2C+1)}{8A-4(B+C)}$$ and ...
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How to solve ordinary differential inequations with vector variables?

Given $a\in\mathcal{R}_+^d$ and $s\in\mathcal{R}^d$,we wanna a function f(.) which maps s to a vector $f=\begin{bmatrix}f(s_1),\cdots,f(s_d)\end{bmatrix}^T$ and satisify the following inequation. ...
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Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
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38 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
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How prove this inequality with $\sqrt{a_{1}+\sqrt[3]{a_{2}+\sqrt[4]{a_{3}+\cdots+\sqrt[n+1]{a_{n}}}}}$

let $a_{i}>0,i=1,2,\cdots,n$, show that $$\sqrt{a_{1}+\sqrt[3]{a_{2}+\sqrt[4]{a_{3}+\cdots+\sqrt[n+1]{a_{n}}}}}\ge \sqrt[\displaystyle{(2!+3!+4!+\cdots+(n+1)!)}]{a_{1}a_{2}\cdots ...
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Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
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How find this minimum of the $|PA_{1}|+|PA_{2}|+|PA_{3}|+\cdots+|PA_{n}|$

Question: give the $n$ point $$A_{1}(x_{1},y_{1}),A_{2}(x_{2},y_{2}),A_{3}(x_{3},y_{3}),A_{4}(x_{4},y_{4}),\cdots,A_{n}(x_{n},y_{n}),x_{i}\in R,y_{i}>0$$ Find a ponit $P(x,0)$,such ...
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Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
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How to prove this inequality $b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$

let $a,b,c>0$,and $$abc=1$$ show that $$b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$$ since $$b(a-1)(c-1)=b(ac-a-c+1)=abc-ab-bc+b=1-ab-bc+b$$ so we only prove $$3-2(ab+bc+ac)+a+b+c\le 0 $$ oh,this ...
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Inequality among trigonometric sums of normal random variables

This is an inequality used in a proof which I do not know how to prove. $$\left(\sum_{k = 2^j +1}^{2^{j+1}} \frac{\sin(k\pi t)}{k}G_k\right)^2 \leq \left|\sum_{k = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi ...
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Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
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Is there an effective bound known for the coefficients of half integer weight cusp forms?

If $f(z)=\sum a_n q^n$ is a cusp form (of integer weight) normalized so that $a_1=1$, we have the inequality $$\vert a_n \vert \leq d(n) n^{(k-1)/2},$$ known as the Deligne bound (in which $d(n)$ ...
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109 views

Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...
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131 views

Practicing the arithmetic-geometric means inequality

I am struggling with learning the AM-GM Inequality that is considered vital to know for math olympiads, contests, etc. I just don't really know when to use it, when it is necessary to use, the purpose ...
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76 views

A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
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75 views

Almost Jensen's Inequality

Let $a,b$ and $c$ three positive reals numbers such that $abc=1$. Define the function $f$ by $f(x)=\frac{^1}{1+(n-1)x^n}$ where $n$ is a positive integer. Prove that ...
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Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
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40 views

Decomposition of polynomials and inequality

This was asked in comment here by @23rd : If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
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100 views

Inequality with size of sets

Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1}+p_2+\cdots+p_k= 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ ...
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Do these inequalities make sense?

I have two sets of inequalities and i just want to know if they are correct. The parameters $\mu, K, d_1, \sigma_1,\sigma_2$ and dependent variables $H,F$are positive. Also $\sigma_2>\sigma_1$. ...
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32 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
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Can we prove this inequality in another way?

As explained here, I've managed to prove the following inequality: $\sigma(n)\geq\sqrt n(d(n)-2)+n+1$. This can be proved easily in two cases (one for $n$ being a perfect square and one for otherwise) ...