Questions on proving, manipulating and applying inequalities.

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15 views

Plot complex inequalities $|z^2| > Im(z)^2$

I need to draw the set of all complex numbers, which satisfy the following inequality: $|z^2| > Im(z)^2$ This is what I've already done: $|z^2| > Im(z)^2$ $|z|^2 > Im(z)^2$ - use $z = a ...
2
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28 views

How does this inequality imply this one?

I am having a little trouble understanding this part of a proof. There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$ Now, $\text{J}_0 = \frac{\pi ^3}{24} $ The part of ...
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16 views

Requesting for details of Grun's result on odd perfect numbers

This post is an offshoot of this earlier MSE question. According to this Wikipedia page: Grun $1952$ proved that the smallest prime factor $p_1$ (of an odd perfect number $N$) is $< ...
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47 views

Is this trigonometric expression always strictly positive?

Let us define a function $f(k,n)$ by \begin{equation} f(k,n)=n \left (\cos\frac{k\pi}{n}\right) \left(1-\cos\frac{k\pi}{n}\right) - \sin \frac{k\pi}{n} \end{equation} Where $\frac{k}{n}$ is ...
2
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38 views

Cases in which a certain inequality holds true.

I previously asked this question on this forum, and have been demonstrated counterexamples to the claim that $|a| > |b|$ implies $\big|\frac{b+b^{2}}{a+a^{2}}\big| < 1$, which I had previously ...
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32 views

Finite sequence created by reducing $n$ with each prime under $n$ ends in $0$?

Given $n$ a fixed integer we constuct the following sequence: $a_0=n$, $a_i=\lfloor \frac{a_{i-1}(p_i-1)}{p_i}\rfloor$. For what values of $n$ do we have $a_{\pi(n)}=0$? Computer calculation shows ...
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58 views

Elementary-Looking Inequality on n Complex Numbers

Let $z_1,z_2,\ldots,z_n$ be complex numbers. Is it true that $\displaystyle\sum_{1\le i,j\le n} |z_i+z_j| \ge \displaystyle\sum_{1\le i,j\le n} |z_i-z_j|$? I know the inequality holds for reals and ...
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27 views

Choosing three integers to satisfy an equation under a specific condition

Find three integers $(a,b,c)$ such that: $x*a + y*b + z*c = a + b$ only when $x = 1, y = 1, z = 0$ where $x, y$ and $z$ can be chosen as any non-negative integers. For example, choosing $a = 1$; $b = ...
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58 views

I am trying to show an inequality involving the product of three inner product terms

Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as: $$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$ where $\rho$ is a real number. I ...
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35 views

Prove that $(n-1)!S_m\geq (n-m)!m!P_m.$

If $a_1, a_2,\cdots a_n$ be all positive rationals such that $S_n=a_1^m+a_2^m+\cdots +a_n^m$, $P_m=\sum a_1a_2\cdots a_m$ (the sum of products m taken m at a time). Prove that $$(n-1)!S_m\geq ...
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36 views

Proving $n\cdot\bigg(\sum_{i=1}^{n}|x_{i}|^{2}\bigg) \leqslant (n\cdot|x_{n}|)^{2}$

May this be proved by induction? Let $x \in \mathcal{R}^{n}$, where $|x_{n}| \geqslant |x_{i}|$, $i \neq n$. In other words, $|x_{n}|$ is the maximal element of $x$. Then, $$ ...
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50 views

Inequality with three variables

Let $a,b,c\ge 0$,show that $$\sqrt{a^3+2}+\sqrt{b^3+2}+\sqrt{c^3+2}\ge \sqrt{\dfrac{9+3\sqrt{3}}{2}(a^2+b^2+c^2)}$$
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33 views

Doob's $L^p$ inequality - Case $p = 1$

I have found in the wikipedia page following generalisation of Doob's so-called $L^p$ inequality, for general nonnegative submartinagles $X_s$: $$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + ...
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46 views

Solving Higher Order Polynomial Inequalities in Two Variables

Could someone please point me to a solver (software) and also some techniques to solve equations of the kind shown below? $$ x^{3}y+ay^{3}+by^{2}+cy+dx+exy+fxy^{2}\geq0 $$ Here, $x,y$ are the ...
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30 views

Inequality of expansion of $\| \ \ \|$

The following is from a proof of a paper Distributed subgradient Methods for Multi-Agent Optimization, Nedic & Ozdaglar - Lemma 5 Note: ...
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46 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, ...
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12 views

Prove this inequality $|t_{1}-t_{2}|<2mp\left[1-(mp)^{-\frac{p}{p+1}}\right]$

Let $$f(x)=x+\dfrac{m}{x^p},m,p>0$$ if $t_{1},t_{2}$ such $$f(t_{1})=f(t_{2})=mp,t_{1}\neq t_{2}$$ show that $$|t_{1}-t_{2}|<2mp\left[1-(mp)^{-\frac{p}{p+1}}\right]$$
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33 views

Problem related matrix and counting

I have a following matrix related problem: Let $F$ be a $n \times n$ discrete Fourier matrix defined as $F_{j,k} = \frac{1}{\sqrt{n}}exp{(\frac{i 2\pi jk}{n})}$, for $0 \leq j,k \leq n$, where $i^2 ...
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39 views

Khintchine inequality (question about Holder inequality)

I'm reading a proof of a Khintchine inequality : Let $(r_{1}, \dots , r_{n})$ be iid random variables with $P(r_{i} = \pm1) = \frac{1}{2}$. Let $f = \sum\limits_{j=1}^{n}a_{j}r_{j}$, where ...
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367 views

Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...
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27 views

Dospinescu's Inequality

Consider $a_1,\dots, a_n$ be positive real numbers such that $a_1 a_2 \dots a_n = 1$. Prove that $$n^n \prod_{i=1}^n \bigg ( 1 + a_i^n \bigg ) \geq \bigg( \sum_{i=1}^n a_i + \sum_{i=1}^n \frac 1 a_i ...
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115 views

(algorithms) Show that in any base b >= 2, the sum of any three single-digit numbers is at most two digits long

So, I'd like someone to review my 'proof' and pick on it for incompleteness, and state how it could be improved. The question (reviewing algorithms) asks, "show that in any base b>=2, the sum of any ...
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32 views

A inequality of area in Complex analysis

Let $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}a_{n}z^n$ is a univalent holomorphic function in$B(0,1)\setminus\{0\}$,How to prove that$$\sum_{n=1}^{\infty}n|a_{n}|^2\leq1.$$
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29 views

Doubt about inequalities

My original question is if $f\in \mathcal{S}_{\alpha_1}^{\beta_1},\: g \in \mathcal{S}_{\alpha_2}^{\beta_2}$, where does $(f\cdot g) (x)=f(x)g(x)$ belong ? where $\mathcal{S}_\alpha^\beta$ is defined ...
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37 views

Around an inequality

I have a very general question, hopefully not too general. Assume that we have real numbers $a_{ij}, b_{ij}$ $(1 \leq i, \: j \leq n)$ such that $-1 \leq a_{ij}, b_{ij} \leq 1$ for all $i,j,$ for ...
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56 views

Is the following inequality involving matrix exponential true?

Let $X$ and $L$ be real positive definite matrices. $$\operatorname{Trace}(X^{-1}(X - e^{\log(X) - L})^2) \leq \operatorname{Trace}(XL^2)$$ where the exponential and the log are matrix exponential ...
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44 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 $[a,b]$ ...
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104 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
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63 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 ...
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38 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, I try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
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24 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}|| ...
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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 ...
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55 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 ...
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49 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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21 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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28 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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36 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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81 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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202 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. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
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60 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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41 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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56 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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46 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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22 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 $x\in [0,x_0]$ ...
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91 views

inequality about characteristic function

Let $X$ be a random variable with density $f(x)=|x|^{-3}1_{|x|\ge1}$ and $\phi_{X}(t)=E[e^{itX}]$. Show that $\forall t\in[-1,1] $ $$|\phi_{X}(t)-1-t^2log|t||\le3t^2$$ I noticed that $E[X]=0$, so ...
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81 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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123 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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0answers
44 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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0answers
23 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 ...
2
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0answers
105 views

Case of equality in Bernoulli's inequality

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