Questions on proving, manipulating and applying inequalities.

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3
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25 views

inequality involving lifts of a positive oriented homeomorphism of the circle

Let $\pi: \mathbb R \to S^1$ be the natural projection and let $f:S^1 \to S^1$ be a positive oriented homeomorphism. We say that $F: \mathbb R \to \mathbb R$ is a lift of $f$ is $\pi \circ F = f \circ ...
3
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73 views

Howto prove that $\sum_{cyc}\cos\frac{A}{2}\cos\frac{B}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$

let $ABC$ is a triangle with inradius $r$ and circumradius $R$. Show that $$\cos\frac{A}{2}\cos\frac{B}{2}+\cos\frac{C}{2}\cos\frac{B}{2}+\cos\frac{A}{2}\cos\frac{C}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{...
3
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75 views

Prove this inequality for all integers $m>2$

Prove this inequality for all integers $m$: $$∑_{n=2}^{m} \frac{1-n^{2α-1}}{n^{\alpha}} > \frac{1-(m+1)^{2α-1}}{(m+1)^{\alpha}}$$ for all $0<α<1/2$ and $m>2$.
3
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170 views

How prove this sum equality?$\le\prod_{i=1}^{n}\left(\sum_{k=1}^{m}a_{k,i}\right)$

consider this matrix $$\begin{bmatrix} a_{1,1}&a_{1,2}&\cdots,&a_{1,n}\\ a_{21}&a_{2,2}&\cdots&a_{2,n}\\ \cdots&\cdots&\cdots&\cdots\\ a_{m,1}&a_{m,2}&\...
3
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96 views

Find the minimum value of: $P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$

Let $a,b,c\ge0$ such that: $(a+b)(b+c)(c+a)=1$. Find the minimum value of: $$P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$$. I've tried many things but all failed. Please ...
3
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92 views

A challenging non homogenous fractional inequality.

The following problem is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a ...
3
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236 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and $\Omega(N)$...
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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277 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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330 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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696 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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215 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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299 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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24 views

On some iterated inequalities and $x \geq 5$

Let $x \in \mathbb{N}$. Suppose that I have a function $f:\mathbb{N}\rightarrow\mathbb{Q}$, with initial bounds $$2 - \frac{2}{x_0} < f(x_0) = \frac{2{x_0}}{x_0 + 1} \leq 2 - \frac{5}{3x_0}.$$ ...
2
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19 views

Upper estimate of function such that $\int |x|^2 f(x) dx<\infty$

Let $f$ be a non-negative function on $\mathbb R^d$ satisfying the following: (1) There exists a non-increasing function $g$ on $(0,\infty)$ such that (1-i) $ C_1^{-1} g(|x|) \le f(x) \le C_1 g(|x|)...
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44 views

If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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56 views

olympiad-type inequality

Prove that for any $x_1,\dots,x_n>0$ $$ {\root{n}\of{\prod _{k=1}^{n}\ \sum_{t=1}^{k}\ \frac{1}{t^2\cdot\sqrt[t]{x_1\cdot\ldots\cdot x_t}} }} \ \cdot\ \sum _{k=1}^{n}\frac{\sum_{j=1}^{k}\sum_{i=1}^...
2
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53 views

Generalization of Jensen's inequality to multivariate functions

Is there a generalization of Jensen's inequality for convex multivariate functions? By convex, let's say $f$ is a multivariate function defined on the convex set $A$, and for all $x,y \in A$ and $\...
2
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76 views

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $[px]+[py]\ge [qx+y]+[x+qy]$

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $$\lfloor px \rfloor + \lfloor py\rfloor \ge \lfloor qx+y\rfloor+\lfloor x+qy \rfloor$$ I used $x\...
2
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48 views

Partitioning a set of integers (with Alice and Bob)

Let $ d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} $ (not necessarily distinct) be given. Define $ D:=\operatorname{lcm}(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $. (1) Alice claims that whenever $ \...
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35 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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31 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 $C_{n}$? $$...
2
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41 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)^...
2
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40 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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67 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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55 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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60 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 ...
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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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123 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(...
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35 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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36 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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0answers
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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49 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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0answers
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
votes
0answers
33 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 ...
2
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0answers
63 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 ...
2
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0answers
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
votes
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 ...
2
votes
0answers
37 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!...
2
votes
0answers
37 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(\...
2
votes
0answers
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
votes
0answers
47 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
votes
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
votes
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 ...