Questions on proving, manipulating and applying inequalities.

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3
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114 views

Inequality of covariances between a bivariate normal vector and its indicator functions

Why holds for a standardized bivariate normal vector $Z:=(Z_1,Z_2)$ that \begin{equation} |\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|? \end{equation} ...
3
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276 views

Chebyshev and Markov inequalities

Chebyshev inequality: Let $(\mathcal{X},\mathcal{A},\mu)$ be a measurable space, $f$ a non-negative measurable function defined on $\mathcal{X}$. Then, $$\mu([f>c]) \le \frac{1}{c^p} \int_{\mathcal{...
3
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84 views

An inequality with Gamma Function

Consider the following function for any $a, b > 0$ $$ \ g\left( a,b\right) = \frac{% 3\Gamma \left( 3b+1\right) }{\Gamma \left( \frac{1}{a}+3b+1\right) }-\frac{% 5\Gamma \left( 2b+1\right) }{\...
3
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100 views

Geometrical Inequality

Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals $AC$ and $BD$ intersects at $E$. If the shortest height of the triangle $ACD$ equals the radius of the incircle of the triangle $ABE$...
3
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112 views

a simpler proof for an inequality in probability

I am trying to show that for any two real $X,Y$ iid there holds that $$ P(|X - Y| \le 2) \leq 3P(|X - Y| \le 1) $$ I am getting nowhere with this and so I did some digging and found the following ...
3
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328 views

Trigonometric inequality proof

Can anyone help me in proving that $$\cos\theta > \frac{\left(x^a\cos\theta-(x-1\right)^a\cos\frac{\ln x\theta}{\ln(x-1)})\cos(\theta+\gamma)}{\cos\gamma},$$ where $a<1$, $x\in \mathbb{N}$, and $...
3
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687 views

A special case of Young's inequality for convolutions

The problem: Suppose $f,g\in L^1(\mathbb{R})$. Let $x\in \mathbb{R}$ and $\phi_x(y) = f(y)g(x-y)$. Show that for almost all $x$, $\phi_x$ is integrable. For such $x$ let $\psi(x) = \int_{-\infty}^\...
3
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208 views

Inequality of ODE solutions

Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and $v(...
3
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297 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq HM(\mathbf{x}),...
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19 views

Let a, b, c>0, such that a+b+c=1, prove that:

Let a, b, c>0, such that a+b+c=1, prove that: $$\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$$
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25 views

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ...
2
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22 views

Finding a maximum with some constraints

I would like to maximize the term $ l_1b_1+l_2b_2+l_3b_3-2 $ such that the following conditions hold: $ 1>l_1>l_2>l_3>0 $, $ l_1,l_2,l_3 \in \mathbb{Q} $, $ b_1,b_2,b_3 \in \mathbb{N} $...
2
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30 views

Find the best constant $C_{n}$ such this complex inequality

nd we can consider this problem In general? if $|z_{1}|=|z_{2}|=\cdots=|z_{n}|=1$ if there exist complex $z(|z|=1)$ such $$\sum_{i=1}^{n}\dfrac{1}{||z-z_{i}||^2}\le C_{n}$$ find the best $C_{n}$? $$...
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40 views

Prove this by inequality with four variables inequality

Let $a,b,c,d>0$ show that $$\color{blue}{\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)^2\ge 4(a^2+b^2+c^2+d^2)+\dfrac{8}{3}[(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^...
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102 views

Prove $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$

$x,y,z >0$, prove $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$$ This inequality is easier than the other one. Previously, I learned ...
2
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39 views

Prove $\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$ for positive $a,b,c$

For $a,b,c >0$, prove $$\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$$ My notation $$\sum_{cyc}a^2b= a^2b+b^2c+c^2a$$ What I try: 1. Using C-S ...
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66 views

Prove $\sum_{cyc}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$ for positive $x,y,z$

$x,y,z > 0$, prove $$\sum_{\text{cyc}}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$$ While this inequality can be proved by brute force, the elegant ...
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52 views

Generalized weighted mean inequality

Let ${p_{1}},{p_{2}},\ldots,{p_{n}}$ and ${a_{1}},{a_{2}},\ldots,{a_{n}}$ be positive real numbers and let $r$ be a real number. Then for $r\ne0$ , we define ${M_{r}}(a,p)={\left({\...
2
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97 views

$\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$

Let $a$, $b$ and $c$ be positive numbers. Prove that: $$\frac{a^3}{13a^2+5b^2}+\frac{b^3}{13b^2+5c^2}+\frac{c^3}{13c^2+5a^2}\geq\frac{a+b+c}{18}$$ A big problem around $(0.785, 1.25, 1.861)$. In ...
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58 views

Inequality involving fourth powers .

I have been into inequalities lately and I am stuck with this. I used a famous inequality at first $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge 3 (\frac{a^4+b^4+c^4}{3})^{\frac{1}{4}}$. From this I ...
2
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41 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! \...
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122 views

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(...
2
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34 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
2
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35 views

$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}$ inequality is used to prove the theorem

In the book Randomized Algorithms from Motwani and Raghavan, it is stated in page 44 that $$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}.$$ ...
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35 views

Equality in Hardy's inequality via Hölder's

I'm working on Exercise 3.14 in Rudin's Real and Complex Analysis. I was able to answer part (a): that for real $p$ satisfying $1<p<\infty$, for every function $f$ in $L^p(0,\infty)$, when $F$ ...
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16 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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29 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 $< \frac{2\...
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48 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 ...
2
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0answers
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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62 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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29 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 = ...
2
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0answers
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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36 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 (n-m)!m!...
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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, $$ n\cdot\bigg(\...
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57 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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34 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 + E[X_T\log^...
2
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0answers
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 ...
2
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0answers
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: $y(k+1)=y(...
2
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0answers
47 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 $n$, the ...
2
votes
0answers
14 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 =...
2
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0answers
44 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 $a_{...
2
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0answers
422 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 $f$ ...
2
votes
0answers
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 \...
2
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0answers
119 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 ...
2
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0answers
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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0answers
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 ...
2
votes
0answers
71 views

Inequality with $x^2+y^2+z^2+xyz=4$ condition

For $x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove that $$ 4(xy+yz+zx-xyz) \geqslant (x^2y+z)(y^2z+x)(z^2x+y)$$ Observations The condition $x^2+y^2+z^2+xyz=4$ is special. One can use the ...