Questions on proving, manipulating and applying inequalities.

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2
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41 views

Question about the assumption of a version of Grönwall's inequality.

According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following: Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ ...
2
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58 views

Is my logic on general Proof-Solving techniques correct?

I've just recently started working through proofs for what's really the first time in my life. Throughout high school, and thus far in college I've never really had to prove things too often, and if I ...
2
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23 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
2
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63 views

Upperbound for $\sum_{i=1}^n\frac{1}{x_i^2}$?

Suppose that $x_i>0$, $i=1,\ldots,n$. I'm looking for an upperbound (doesn't have to be particularly tight) of $\sum_{i=1}^n\frac{1}{x_i^2}$ in terms of some symmetric function of ...
2
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41 views

One-sided Bound on Sum of Fourth Moments

I'm interested in methods for proving one-sided bounds of the form $$ \mathbb{P}[\frac{1}{n}\sum_{i=1}^n X^4_i \geq 3+t]\leq Ce^{-nt} $$ where $X_i$ are standard normal random variables. I've run a ...
2
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48 views

Is this system of inequalities (and equality) tractable?

I have some real parameters here. The $\mu_i$ - for $i=1,2,3,4,5$ - are 'convex coefficents' in that $\mu_i\geq 0$ and $\sum_{i}\mu_i=1$. The $x$ and $z$ are such that $x^2+z^2\leq 1$. The ...
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20 views

An inequality involving an exponential rate of sum

I'm having trouble understanding the conclusion in the proof of Cramér's Theorem in $\mathbb{R}^d$ in the book by Dembo/Zeitouni: We have the following: $\delta>0$ is fixed, $B_{y,\delta}$ is the ...
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27 views

Optimization by Symmetry?

Let $$f(x,y,a,b) := \frac{xa+yb}{\sqrt{xa^2+yb^2}},$$ where $x,y,a,b$ are all positive. Define $$g(a,b) = \min_{x+y=1,\,x,y\ge 0}f(x,y,a,b).$$ How would one solve for $g(a,b)$? I have solved this by ...
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33 views

Schwarz Inequality of function from upper half plane to disc

So I've been working on this problem and I have everything nailed down (I think) except for the very end. In particular I get a bound, but I can't seem to reduce it down to the one the question is ...
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74 views

Is the Schwarz inequality a special case of the Cauchy-Schwarz inequality?

Given two vectors $\mathbf{x},\mathbf{y}$ in $\mathbb{R}^n$, we all know that:$$\left | \mathbf{x}\cdot\mathbf{y} \right | \le \left \| \mathbf{x} \right \| \cdot\left \| \mathbf{y} \right \|$$ ...
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197 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
2
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50 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
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40 views

Where is this inequality coming from?

It's probably simple but I'm not sure why I'm not seeing it. The inequality is from a paper: $$\begin{align*} \sum_{i=1}^4 \rho_i (x_i-1)(1-\sum_{j=1}^4 \alpha_{ij}x_j) &\leq\begin{split} ...
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54 views

asking a way to prove an inequality

Assume $\Omega$ is a bounded smooth domain in $\mathbb R^N $ with $N \ge 5 $ and $u \in C^2(\Omega)$ . I want to proof $$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; ...
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44 views

Prove $\frac{1}{\pi^2}\int_0^x \left(\sin \pi t\right)^2\left[\frac{1}{(t-a)^2}+\frac{1}{(t+a)^2}\right]dt\geq \frac{x-a}{1+x-a}$

Prove that $$\frac{1}{\pi^2}\int_0^x \left(\sin \pi t\right)^2\left[\frac{1}{(t-a)^2}+\frac{1}{(t+a)^2}\right]dt\geq \frac{x-a}{1+x-a}$$ for every $x\geq a>0$. I do not know where to start! Any ...
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20 views

Confused about Landau-notation and inequality

Let $f$ be a real valued function and $|f(x)| \le x^2\cdot C + o(x^3)$ as $x\to 0$, where $C \ge 0$ is a constant independent of x. Is it true that there is a $x_0$ such that for all $x\in [0,x_0]$ ...
2
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69 views

Spectral norm bound for difference of inverse matrices [modified]

if A is positive semidefinite and B,C are positive definite matrices, can you bound the spectral norm $\lVert B(I+AB)^{-1}AC - B(I+AC)^{-1}AC\rVert $ by something like $\lVert B\rVert \lVert B - ...
2
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114 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ an $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
2
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31 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i) \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
2
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22 views

$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$

Let $c_i\in\mathbb R$, $a_i\geq0$ with $\sum_{i=1}^n a_i=1$, prove $$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$$ This inequality comes from there, when $X$ is ...
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93 views

Case of equality in Bernoulli's inequality

How can I prove that the following equality holds only for $x=0$? $$\binom{n}{2}x^2 +\cdots+ \binom{n}{n}x^n=0\text{ when }x\gt-1\text{ and }n\gt1$$
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40 views

Equality or inequality

Sorry to make this foolish query. Does the following inequality is necessary? (it is correct to use less or equal to), then the expressions on the left and right are equal or unequal it is actually ...
2
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48 views

How prove $\frac{a^2}{(a+b)^2}+\frac{b^2}{(b+c)^2}+\frac{c^2}{(c+a)^2} \ge \frac{3}{4}+\frac{(a-b)(b-c)(a-c)}{(a+b+c)^3-3abc} $?

Let $a \ge b \ge c >0$ . How prove $\frac{a^2}{(a+b)^2}+\frac{b^2}{(b+c)^2}+\frac{c^2}{(c+a)^2} \ge \frac{3}{4}+\frac{(a-b)(b-c)(a-c)}{(a+b+c)^3-3abc} $? Maby simple way?
2
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29 views

Obtain an inequality of real numbers

Let $x,y> 0$ be real numbers such that $x>y$. Let $\alpha \in (0,1/2)$ be a parameter then I obtained the following inequality: $$y^{-\alpha} - x^{-\alpha} \leq C ...
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23 views

Determining asymptotics of a function given a series of difference-like inequalities

I have a function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and I know it satisfies the following properties. $f(x) \leq \frac{\log{\sqrt{2}}}{2x}$ and for all $A \geq 1$ and $B \geq ...
2
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48 views

On the second part of solution of a question due to Erdos

Problem. Let $a_1<a_2<\dotsb<a_n\le 2n$ be a sequence of positive integers. Then $$ \min [a_i,a_j]\le 6\left(\Big[\frac n2\Big]+1\right), $$ where $[a_i,a_j]$ denotes the least ...
2
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46 views

Finding maxima of a 3-variable function.

Let $x,y,z$ be positive real number satisfy $x+y+z=3$ Find the maximum value of $P=\frac{2}{3+xy+yz+zx}+(\frac{xyz}{(x+1)(y+1)(z+1)})^\frac{1}{3}$
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177 views

Improvement of an Inequality

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
2
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32 views

On the existence of a certain sequence of positive numbers II

I wish to find a sequence of strictly positive real numbers $(a_1, a_2, \dots)$, such that $$ \sum_{k = 1}^\infty \frac{a_k}{k} < \infty $$ and such that for all $m, n \in \{1, 2, \dots\}$ with $m ...
2
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150 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions $\forall i, a_i \geq 0, ...
2
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28 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
2
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45 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
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43 views

Why is the bold text true??

Ok I have stared at this for nearly 30 minutes now, and can't figure out why the bold text is true. Problem: If $z \in \mathbb{C}$ and $\mathrm{Re}(z^n) \ge 0$ for $n \in \mathbb{N}$, show that $z ...
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31 views

Finding feasible solution of inequalities in math software

I have a Math problem where I have some true statements, and I want to know if there is a feasible solution to an equation. I would like to know how to do that in either Matlab or Mathematica. The ...
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44 views

Truth of an inequality involving differentials

Is the following inequality true? $$ s\frac{\partial \frac{\partial f(s,t)}{\partial s}}{\partial t}-\frac{\partial f(s,t)}{\partial t}>0 $$ Given that $f(s,t)$ is a monotonically-decreasing ...
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113 views

Bound for variance of maximum of normal random variables

Suppose that $(X_1,\ldots,X_n)=\mathbf{X}\sim N(\mathbf{0},\Sigma)$ is an $n$-dimensional normal random vector. I want to show the bound $$ \text{Var}\left(\max_{i\leq n} X_i\right)\leq \max_{i\leq n} ...
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53 views

Showing $\frac{1+c}{a+b}\leq \varphi$, when $\frac{1}{1+b}\leq a\leq 1$, $\frac{c^2}{a+c}\leq b\leq 1$ and $0\leq c\leq 1$

The title pretty much says it all. Let $\varphi\triangleq\frac{1+\sqrt 5}{2}$ be the golden ratio. Let $a,b,c$ be some non-negative numbers such that: $\frac{1}{1+b}\leq a\leq 1$ ...
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120 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...
2
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120 views

Proving the inequality $\frac{\mathrm{arccot} 2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$

I have come across the following inequality in my studies $$\frac{\text{arccot}2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$$ where ...
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45 views

Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ ...
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49 views

How prove $ \frac{2}{\sqrt3}F \geq s-1 $ for convex quadrilateral?

Let $Q$ be any convex quadrilateral of area $F$ and semiperimeter $s$. Suppose that length of any diagonal of $Q$ $ \geq$ length of any side of $Q$ $\geq 1$ How prove $ \frac{2}{\sqrt3}F \geq s-1 ...
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56 views

Reference for an inequality between $|x+y|^p$, $|x|^p$, $|y|^p$, and $|x-y|^p$

I am interested in an inequality for real numbers $x,y$ and $1<p<\infty$ that it should say something like $$ |x+y|^p -|x|^p - |y|^p \leq (1-2^{1-p})|x-y|^p. $$ Is this inequality (or ...
2
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72 views

Two-leg games in Elo rating for football teams

Do you know Elo rating for association football? It is a numerical estimation of strength of football clubs using simple mathematical formula based past results allowing predictions for the future. ...
2
votes
0answers
91 views

Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes ...
2
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0answers
61 views

System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
2
votes
0answers
73 views

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} ...
2
votes
0answers
40 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, ...
2
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0answers
61 views

On the average length of the Steiner net for $n$ randomly chosen points in the unit square

$n$ points are randomly chosen in the unit square with respect to the uniform measure. What is the average length $L$ of the associated Steiner net (tree of minimum length through each of the $n$ ...
2
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90 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
2
votes
0answers
100 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...