Questions on proving and manipulating inequalities.

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“Massaging” inequalities to prove them (esp. in contest math like the IMO/Putnam)?

What's the contest inequality solving technique where you do something like representing each side as the function of some sequence and replacing the max/min terms of the sequence with their average, ...
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32 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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59 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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50 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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52 views

“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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33 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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112 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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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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27 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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41 views

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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55 views

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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19 views

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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132 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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24 views

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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29 views

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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124 views

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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86 views

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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42 views

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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55 views

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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77 views

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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106 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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125 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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72 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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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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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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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) ...
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Radical Inequality related to three variables

Prove the following. $$\sum_\text{cyclic}\sqrt[4]{\dfrac{(a^{2}+b^{2})(a^{2}-ab+b^{2})}{2}}\leq\dfrac{2}{3}\left(\sum_\text{cyclic}\dfrac{1}{a+b}\right)\left(\sum_\text{cyclic}a^{2}\right)$$
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max value of difference of square roots of sums

Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the ...
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154 views

An number theoretic inequality

Prove this inequality : $$\displaystyle \prod_{i\le \left\lfloor{\frac{n-1}{2}}\right\rfloor}\left\lfloor{\frac{\left\lfloor{\frac{n-1}{2}}\right\rfloor}{i}}\right\rfloor\le ...
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39 views

How find this maximum $\sum_{i=1}^{2000}\left(\frac{x^{2000}_{i}}{\sum_{j=1}^{2000}x^{3999}_{j}- i\cdot x^{3999}_{i}+2000}\right)$

Question: let $x_{1},x_{2},\cdots,x_{2000}$ be real numbers,and such $x_{i}\in [0,1],i=1,2,\cdots,2000$.and define $$F_{i}=\dfrac{x^{2000}_{i}}{\displaystyle\sum_{j=1}^{2000}x^{3999}_{j}- ...
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55 views

Is there an inequality such that $\sum_{k=1}^{n}a_{k}b_{k}\leq \frac{1}{n}(\sum_{k=1}^{n}a_{k})(\sum_{k=1}^{n}b_{k})$?

Is there an inequality such that $$\sum_{k=1}^{n}a_{k}b_{k}\leq \frac{1}{n}(\sum_{k=1}^{n}a_{k})(\sum_{k=1}^{n}b_{k})$$ with some restrictions on $a_{k}$ and $b_{k}$?
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51 views

Prove an inequality with two sigma notations

I am stuck at this inequality question that I cannot solve. The question writes: Given a list of numbers k0, k1, k2, ...,kn that k1+k2+...+kn=0, and k0=0 Prove: I find the (j-i) part extremely ...
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55 views

hard inequalities

I have to find real $x$ that satisfy the equation: $\dfrac{x^7}{7} = 1+10^{1/7}x(x^2-10^{1/7})^2$ I saw that the way is to look for solution of the form: $x = a^{1/7}+b^{1/7}$. my question is: how ...
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39 views

What conditions must satisfiy a positively skewed density function to ensure that median is greater than mean

We are collecting environmental Air Quality data. When we validate data, we always plot ECDF and compute basic statistics and percentiles. Our experimental distributions are far away from normality. ...
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25 views

Finding a derivative through the definition

Let $f$ be a function of domain $\mathcal D$ and codomain $\mathcal C$, both subsets of $\mathbb R$, and $\mathcal D_{\mathrm{cl}}$ the set of cluster points of $\mathcal D$. I want to find $f'$ by ...
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66 views

Eigenvalue Inequality Involving Hermitian Positive semi-definite matrices

I am trying to determine if an inequality holds for Hermitian matrices $A$,$B$, and $C$ of the same dimension. Let $A$ be positive definite, $B$ and $C$ be positive semi-definite, and $B-C$ positive ...
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40 views

Prove this statement about inequalities

Can someone help to prove this. For $n$ and $\{a_{11},\dots,a_{nn}\}$, if we know that $a_{ij}$ is either $0$ or $1$ or $-1$, and further assume that the following inequality system on $\{b_n|b_n\in ...