Questions on proving, manipulating and applying inequalities.

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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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14 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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47 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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27 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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42 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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39 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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80 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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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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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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50 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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55 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, ...
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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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44 views

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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36 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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43 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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67 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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37 views

$L^2$ 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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43 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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26 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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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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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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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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66 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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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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37 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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23 views

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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127 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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43 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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58 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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22 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 ...