Questions on proving, manipulating and applying inequalities.

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Prove $S$ is an interval of $\mathbb{R}$ $\iff$ it has betweenness

The definition of betweenness is given as: If $x \in S$, $y \in S$, and $x < z < y$, then $z \in S$. However, I don't know the formal definition of interval in $\mathbb{R}$, which makes it ...
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40 views

How to construct solutions for a set of polynomial matrix inequalities

How can one find solutions to the set of (polynomial) matrix inequalities $$M \succ 0,\quad A_i^TMA_i \preceq c\cdot M,\quad\forall i=1,\dots,m$$ where $M=M^T\in\mathbb{R}^{n\times n}$ and $A_i ...
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33 views

A System of Inequalites arising from the Divisors of a Number, Showing Its Non-Solvability

Let $n$ be a natural number. Denote by $d(n)$ the number of divisors of $n$, i.e. with the notation from Wikipedia:Divisor Function we have $d(n) = \sigma_0(n)$. Suppose we have the $d(n) - 2$ ...
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16 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
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16 views

Some inequality related to ODE systems

For $i=1,2$, let $\phi_i:R_+\to R_+$ be a continuous function such that $\phi_i(0)=0$ and define $$\gamma_i(l) := \int_0^l\frac{dm}{\phi_i(m)}.$$ Assume that $(\phi_1,\phi_2)$ satisfy the following ...
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62 views

Prove this inequality to $\min{(a_{1},a_{2},\cdots,a_{n})}$

Let $a_{1},b_{1},a_{2},b_{2},\cdots,a_{n},b_{n}$ be real numbers such that $$\min{(a_{1},a_{2},\cdots,a_{n})}\ge 0$$ show that ...
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25 views

Visualising relations between inequalities and solution criteria.

Is there any intuitive, visual explanation of the following lemma: Lemma: Let $\{ \alpha_{ij} : i = 1, \ldots, m, j = 1,\ldots, n \}$ be an $m \times n$ matrix, $\alpha_i = (\alpha_{i1}, \ldots, ...
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17 views

Is it true that $2\max_{|z|=1} | z^n+ (\sum_{ i=2}^{n-1} a_iz^i ) + a_1 z+a_0| \geq \max_{|z|= 1} |a_1z+a_0|$?

I come across to prove or disprove an inequality which is $$2\max_{|z|=1} | z^n+ (\sum_{ i=2}^{n-1} a_iz^i ) + a_1 z+a_0| \geq \max_{|z|=1} |a_1z+a_0|,$$ where $z\in \mathbb{C}$ and $a_i$ are real. ...
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24 views

Why is this set a subset of its polyhedral approximation - contradicting the gradient inequality?

Say we have a set $C:= \{y\in \mathbb{R}^n : g_i(y) \leq 0, \space i=1,...,m\}$ where $g_i : \mathbb{R}^n \to \mathbb{R}$ are convex and differentiable functions, then we have $\tilde C : = \{y: ...
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89 views

Is the function $f(A)=-\log(tr(A^{-1}))-\log(\det(A))$ convex?

I am trying to show the following function is convex or not $$f(A)=-\log(\text{trace}(A^{-1}))-\log(\det(A)),$$ where $ A$ is positive definite. I know $\text{trace}(A^{-1}), -\log(\cdot)$ and ...
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15 views

Upper bound on function defined by induction involving divisions by two

Let $A=\lbrace (x,y)\in{\mathbb N}^2 | 0<x<y \rbrace$. For positive integers $u,v$ let $\rho(u,v)=(\textsf{min}(u,v),\textsf{max}(u,v))\in A$. It is easy to see that there is a unique map ...
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49 views

Three related inequalities (the first being $2(|a|^p + |b|^p) \leq |a + b|^p + |a - b|^p \leq 2^{p-1}(|a|^p + |b|^p)$)

A friend told me this interesting problem. It should be easy enough, but I cannot figure it out completely. If $a, b \in \mathbb{R}, p \geq 2, \frac{1}{p} + \frac{1}{q} = 1$, then $2(|a|^p + |b|^p) ...
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24 views

Existence of solution for linear matrix inequality?

Suppose $x$ is a $n\times1$ column vector. How to know whether the following matrix inequality has solution or not? $$Ax\leq B$$ where $A$ is a $m\times n$ matrix and $B$ is a $n\times 1$ column ...
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29 views

Leading up to Young's Inequality

I am trying to prove Young's Inequality by considering the function $$h(u) = \frac{u^p}{p} + \frac{C^q}{qu^q}$$ for $C,u>0$ and $p,q >1$. We also require $$\frac{1}{p}+\frac{1}{q}=1$$ so that ...
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21 views

Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
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45 views

Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
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41 views

Is there a name for the inequality $\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$?

Is there a name for the inequality $$\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$$? And does anyone have any nice examples or applications, especially with an economic flavor? The transposed multivariate ...
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26 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
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A maximal inequality on distance to median, so called Lévy's inequality?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.3, ex6) Suppose $X_1,\dotsc,X_n$ are i.i.d. random variables, and $S_j:=X_1+\dotsb+X_j,S_j^0:=S_j-m_0(S_j)$, where $m_0(S_j)$ is a ...
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81 views

Inequality involving a geometric series

Reading a book I can't see how to show the following inequality, $$C\sum_{j,k,l}c_k d_l 2^{-\epsilon|jq+k| -\epsilon |jq+l| -\epsilon |k-l|} \leq \frac{C}{\epsilon} \sum_{k,l} c_kd_l2^{-\epsilon ...
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37 views

Gigliardo-Nirenberg-Sobolev inequality for functions in $W^{k,p}$, without zero trace.

The G-N-S inequality can be stated as follows: Let $U\subset\mathbb R^d$, open bounded, with $C^1$ boundary, then for any $w\in W^{k,p}_0(U)$, $p<d$ $$\|w\|_{L^{p^*}(U)}\le C(d)\|\nabla ...
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30 views

$ \|u(x)+u(x)-x +o(\|x\|^p)\|<r $?

Set $B_r=\{ x\in \mathbb{R}^n : \|x-0\|< r \}$ for any $r>0$. Let $C^p_0(B_r,B_r)$ the set of all smooth functions $u:B_r \to B_r$ of class $C^p$ such that $u(0)=0$. I would like to prove the ...
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39 views

inequality for point on a sphere

Question: Let $a,b,c,d$ be positive real numbers which satisfy $a^2 + b^2 + c^2 + d^2 = 1$. Define $f(a,b,c,d) := \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}$. Show that $$ f(1-a,1-b,1-c,1-d) \geq ...
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55 views

Conditional inequality

Let x,y,z be positive reals with $xy+yz+zx=1$. Prove the inequality $$\sum_{cyc(x,y,z)}\frac {2x(1-x^2)}{(1+x^2)^2} \le \sum_{cyc(x,y,z)} \frac x{1+x^2}.$$ I substituted $x=tan\frac{\theta}2, ...
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67 views

Inequality involving integral of $\Gamma(x)$

The graph of $\frac{e^x}{\Gamma(x+1)}$ is somewhat bell-shaped. I think the proof of the following requires an understanding of the integral of this function that I can't glean from either Mathematica ...
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34 views

Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
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Non-trivial inequality

Equation (1) on page 7 of http://arxiv.org/pdf/1312.7308v1.pdf claims that: $$\frac{1}{t}\log \left(\frac{\log ((1+\epsilon)t)}{\omega} \right) \geq c \Rightarrow t \leq \frac{1}{c} \log \left( ...
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130 views

Prove the Schwarz inequality using $ 2xy \leq x^2 + y^2 $

I'm really bad at analysis and this problem was recommend to me to help me grasp some basics of $\epsilon $ $\delta $ So I'm doing a problem (though it's like 12 pieces) this is I guess the fourth ...
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37 views

Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
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59 views

How prove this the number of ordered $n$-tuples $(\varepsilon_{1},\cdots,\varepsilon_{n})$such this following inequality is $2^{n-100}$

Interesting Question: for any complex numbers $z_{1},z_{2},\cdots,z_{n}$ such $$\begin{cases} |z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2=1\\ |z_{i}|\le\dfrac{1}{10},i=1,2,\cdots,n \end{cases}$$ ...
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45 views

Find a Liapunov function to show asymptotically stable

Consider the system: \begin{cases} \dfrac{dx}{dt} = y \\[12pt] \dfrac{dy}{dt} = -(1+x^{2})\,y-\sin(x) \end{cases} $(0,0)$ is a critical point of this system and I need to show that it is ...
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34 views

Inequality of finite sequences of real numbers .

Is the following inequality true for real numbers $\lambda_{i}$ and $\mu_{i}$ $$\dfrac{\sum_{i=1}^{n}\lambda_{i}\mu_{i}^{2}}{\sum_{i=1}^{n}\lambda_{i}}\times \dfrac{1}{1+\sum_{i=1}^{n}\mu_{i}^{2}} ...
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68 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} ...
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47 views

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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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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27 views

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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63 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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51 views

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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48 views

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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109 views

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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34 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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68 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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54 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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39 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 ...