Questions on proving and manipulating inequalities.

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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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45 views

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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22 views

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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102 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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118 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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67 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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72 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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32 views

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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39 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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98 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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24 views

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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31 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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36 views

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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30 views

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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29 views

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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54 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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35 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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63 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 ...
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40 views

What does it mean by First Order Inequality

As said on the title, sorry for my newbie-ness in the terms.
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Show for any $s \in \mathbb{R}$ that $C_1(1+x^2)^s \le (1+x)^{2s} \le C_2 (1+x^2)^s$.

Let $x \in \mathbb{R}$, then for any $s \in \mathbb{R}$ we must show that there exist $C_1,C_2 \in \mathbb{R}$ such that $$C_1(1+x^2)^s \le (1+x)^{2s} \le C_2 (1+x^2)^s.$$ It seems pretty obvious ...
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Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
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53 views

One-Sided Bivariate Chebyshev Inequality

Let $X$ and $Y$ be random variables with finite means $\mu_X$ and $\mu_Y,$ finite variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho.$ Let $A$ be the event that $X \leq \mu_X + k\sigma_X$ ...
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74 views

Relation between determinant and L1 norm

Recently, I have coped with a problem about the relation between determinant of positive definite matrices and their L1 norm. More specifically, assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two ...
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121 views

Can proof by contradiction and counterexample by used at the same proof?

Here is a part of a theorem: If $\alpha>1$ and $x\ge-1$ then $(1+x)^\alpha \ge 1 + \alpha x$ I was wondering if I could use proof by contradiction and counterexample at the same time. Assume ...
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104 views

Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...
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74 views

How prove this $\sin(2a)+\sin(2b)+\sin(2c)<\dfrac{\pi}{2}+2\sin a\cos b+2\sin b\cos c$

let $0<a<b<c<\dfrac{\pi}{2}$, use the integral inequality show that $$\sin(2a)+\sin(2b)+\sin(2c)<\dfrac{\pi}{2}+2\sin a\cos b+2\sin b\cos c$$ I know this problem can use The area of ...
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46 views

Prove or disprove $\sum_{j=1}^{n}\sum_{i=1}^{n}|a_{i}-a_{j}|^2|b_{ij}|^2\le \cdots$

let $a_{i},b_{ij}\in C,i=1,2,\cdots,n,j=1,2,\cdots,n$,prove or disprove $$\sum_{j=1}^{n}\sum_{i=1}^{n}|a_{i}-a_{j}|^2|b_{ij}|^2\le ...
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53 views

Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
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Schur-concave functions, derivative sign help

To establish some inequality I must prove: $$\dfrac{\partial}{\partial ...
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Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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45 views

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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product of sums

This is a question which has puzzled me for a while. I would be very thankful if somebody can help me with it. Assume you have $S$ rectangles appearing in front of your screen one by one. Each ...
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245 views

Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
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80 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
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Proving that $ (\mathbb E [X^n])^{1/n}\leq (\mathbb E [X^m])^{1/m}$ for $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$

How to prove that for a positive essentially bounded random variable $X$ satisfing $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$ and for any $m,n \in \mathbb N^*$ with $m\geq n$ we have $$ (\mathbb E ...
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94 views

transforming nonlinear matrix inequality to LMI

I faced some nonlinearity in my problems. I need to check a matrix inequality condition in order to check the feasibility of designed controller through a continuous design problem. My problem is that ...
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67 views

Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
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53 views

Proving an inequality involving integral

Let $g: [a,b]\mapsto [0,1]$, with $\int_a^b|g'(t)|^2\,\mathrm{d}t\leq 1$. Suppose $b-a<\delta$, and define $$ \bar{g}=\frac{\int_{a}^{b}g\left(t\right)\,\mathrm{d}t}{b-a} $$ Show for ...
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193 views

Sharpness of the upper bound $(1-x)^n \leq 1 + \frac{nx}{2}$

Here is a known inequality: $$(1-x)^n\leq 1+\frac{nx}{2}\qquad \text{for} \, \frac 1n\leq x\leq 1 $$ I am wondering if there is a better upper bound than this? Thank you.
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30 views

Youngs inequality for any power of X and Y

So I am working on a condition where I want to find the maximum "p" and "q" values for the Youngs inequality shown below . I am looking to find the $p$ and $q$ which upper bounds $x^my^n$ by a ...
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39 views

Showing an inequality relating two Poisson tail-probabilities

In my research, I've discovered that a property that I am interested in is equivalent to an inequality involving two tail-probabilities of the Poisson distribution. I belive this inequality to be ...
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78 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
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29 views

Proving $~\sum_{cyclic}\left(\frac{1}{y^{2}+z^{2}}+\frac{1}{1-yz}\right)\geq 9$

$a$,$b$,$c$ are non-negative real numbers such that $~x^{2}+y^{2}+z^{2}=1$ show that $~\displaystyle\sum_{cyclic}\left(\dfrac{1}{y^{2}+z^{2}}+\dfrac{1}{1-yz}\right)\geq 9$