Questions on proving and manipulating inequalities.

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12
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0answers
265 views

How prove $ \sum (a+b+c+d)\sum\frac{ab+ac+ad+bc+bd+cd}{a+b+c+d}\sum\frac{abc+abd+bcd+cda}{ab+ac+ad+bc+bd+cd}\leq \sum a\sum b\sum c\sum d$

let $a_{i}>0,b_{i}>0,c_{i}>0,d_{i}>0,i=1,2,\cdots,n $ show that ...
10
votes
0answers
240 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
9
votes
0answers
103 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
9
votes
0answers
145 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and ...
8
votes
0answers
139 views
+50

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
8
votes
0answers
105 views

Inequality involving exponential partial sums

Consider the exponential partial sums $E_n(x) = \sum_{i=0}^n \frac{x^i}{i!}$. I want to prove that for all $x \ge 0$: $$2 \frac {E_{n-1}(x)} {E_n(x)} \ge \frac {E_{n}(x)} {E_{n+1}(x)} + \frac ...
8
votes
0answers
49 views

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
8
votes
0answers
379 views

How prove $\sum_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum_{cyc}h_{a}}$

Question: let $\Delta ABC$,and the altitude is $h_{a},h_{b},h_{c}$,where $AB=c,BC=a,AC=b$ and for any $P$ show that $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
7
votes
0answers
107 views

Let $p$=prime and $\sqrt{x}+\sqrt{y}<\sqrt{2p}$

Let $p$ be a fixed odd prime. Let $x,y\in \mathbb{Z}_+$ such that $\sqrt{x}+\sqrt{y}<\sqrt{2p}$. Prove that $$\sqrt{x}+\sqrt{y}\le \sqrt{\frac{p-1}{2}}+\sqrt{\frac{p+1}{2}}.$$ Any ideas at all? ...
7
votes
0answers
111 views

How prove this inequality $(\sum_{k=1}^{3n}a_{k})^3\ge 27n^2\sum_{k=1}^{n}a_{k}a_{n+k}a_{2n+k}$

let $$0\le a_{1}\le a_{2}\le \cdots\le a_{3n}$$ show that $$\left(a_{1}+a_{2}+a_{3}+\cdots+a_{3n-1}+a_{3n}\right)^3\ge 27n^2\sum_{k=1}^{n}a_{k}a_{n+k}a_{2n+k}$$ we know when $n=1$,this is ...
7
votes
0answers
224 views

Proving a geometric inequality without Lagrange multipliers

Let $e=(1,1,\ldots,1)$ be the $n$-dimensional vector consisting only of ones. Let $r=\sqrt{\dfrac{n}{n-1}}$ and $\alpha \in (0,1)$ fixed. Given a vector $x=(x_1,x_2,\ldots,x_n) \in \mathbb R^n$ such ...
7
votes
0answers
174 views

Multivariate polynomial with all coefficients positive

Let $n\geq 3$ be an integer. Consider the following polynomials : $$ f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq ...
7
votes
0answers
487 views

Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
6
votes
0answers
98 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
6
votes
0answers
171 views

Solution set of inequalities in $\mathbb{R}^6$

Let $\theta\in (0,1)$ fixed. We define $A_1$ be the set of all $(a_1,a_2,a_3,a_4,a_5,a_6)\in (0,1)^6$ such that the following conditions hold: \begin{equation} (1) \quad a_2\le a_1, a_4\le a_3, a_6 ...
6
votes
0answers
216 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
6
votes
0answers
632 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
6
votes
0answers
875 views

Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
6
votes
0answers
146 views

Question about an upper bound

Let each of the positive integers $a_1 ,\dots, a_n$ be less than $m$ such that the least common multiple of any two of the positive integers $a_1 ,\dots, a_n$ is greater than the integer $m$. Then ...
6
votes
0answers
255 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
5
votes
0answers
68 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
5
votes
0answers
130 views

Very hard inequality: $\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le k_p \sqrt{a+b+c}.$

Given $p>0$. Find the smallest real number $k_p$ such that the following inequality holds for any non-negative reals $a,b,c$: $$\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le ...
5
votes
0answers
282 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
5
votes
0answers
103 views

Binomial asymptotic.

Is there any "direct" proof of the following asymptotic inequality: let $\alpha\le 1$ and consider $$Q_n(x)=\sum_{k=1}^n\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+k-1)}{k!}x^k$$ Then, $$\int_0^1 ...
5
votes
0answers
159 views

Evaluate integral $ I_s(x) \leq \frac{C}{(\pi(1-2 \alpha s))^{d/2}}\exp\left(\frac{\alpha}{1-2 \alpha s }|x|^2\right) $

For all $ x \in \mathbb{R}^n ,\hspace{5mm} 0 \leq s<t ,\hspace{5mm} t \in \mathbb{R}^+$ $$ I_s(x)=\int_{\mathbb{R}^n}\left|v\left(y \sqrt{2s}+x\right)\right|\exp(-|y|^2) \, \mathrm dy. $$ How we ...
5
votes
0answers
152 views

Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
5
votes
0answers
953 views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
5
votes
0answers
159 views

Bounding function involving Beta functions

Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...
4
votes
0answers
77 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
4
votes
0answers
115 views

Conditions for Trace Inequality Tr( ( A² - B² ) Z) >= 0

Consider the $M \times M$ complex matrices ${\bf A}, {\bf B}, {\bf Z} \succeq {\bf 0}$. We have the relation ${\bf A} \succeq {\bf B} \succ {\bf 0}$, i.e. both are nonsingular. Further assume $\bf Z = ...
4
votes
0answers
135 views

Application of Cauchy-Schwartz with Sobolev norms

I'm working through the problems in the initial value formulation chapter in Wald's General Relativity. A short summary of the problem. I have to show that $$\sup_{x\in A}|f(x)|\le C||f||_{A,k}$$ ...
4
votes
0answers
98 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
4
votes
0answers
56 views

Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's ...
4
votes
0answers
80 views

Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$ Easy to show ...
4
votes
0answers
66 views

How prove this Nice inequality $\sum_\text{cyc}\frac{x^2}{y}\ge 3+\frac{x^4+y^4+z^4-x^2-y^2-z^2}{x^3+y^3+z^3-xyz}$

QUestion: let $x,y,z>0$, show that $$\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 3+\dfrac{x^4+y^4+z^4-x^2-y^2-z^2}{x^3+y^3+z^3-xyz}$$ I know this well know inequality ...
4
votes
0answers
72 views

Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $

I need to prove or disprove the following inequality: $$ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $$ For $x,y,z \in \mathbb R^+$. I found no counter ...
4
votes
0answers
221 views

My favorite proof of the generalized AM-GM inequality: where it came from?

I have already posted (most of) the present question as a (misplaced) answer to a question about understanding a particular proof of the AM-GM inequality. I sincerely hope I am not breaking the code ...
4
votes
0answers
107 views

Diophantine inequality that comes up after Vieta Jumping Hurwitz technique

I am blaming this on Prove the equality EDITTTTT: allowing $x_1 \geq x_2$ and $x_2 \geq x_n,$ I would rather not explain what that was about and the only changes are in $n=3,4,$ already settled. ...
4
votes
0answers
94 views

Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume ...
4
votes
0answers
79 views

How prove this inequality $\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{3}}+\cdots+\frac{x_{n-1}}{x_{n}}+\frac{x_{n}}{x_{1}}-n\le \cdots$

let $x_{i}\in R^{+}$, and such $$x_{1}+x_{2}+\cdots+x_{n}=n$$ show that ...
4
votes
0answers
156 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
4
votes
0answers
232 views

How prove this inequality $\sum_{cyc}\frac{1}{a^5_{1}(a_{2}+2a_{3})^2}\ge\frac{n}{9}$

Question: let $a_{1},a_{2},\cdots,a_{n}>0$,and such $$a_{1}a_{2}\cdot \cdots a_{n}=1$$ prove or disprove: ...
4
votes
0answers
116 views

Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
4
votes
0answers
79 views

Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$ Is this bound (or any other form of hoeffding) tight in ...
4
votes
0answers
70 views

How to prove $\sqrt[n]{n}$ monotone decreases using inequality? there is a hint but I can't

How to prove $\sqrt[n]{n}>\sqrt[n+1]{n+1}$ ,$n\ge 3$monotone decreases by using this hint? I can solve it in other ways but I don't know how to solve it using this hint. hints:consider the ...
4
votes
0answers
234 views

How prove this inequality $a+b+c+d=4$

let $a,b,c,d$ be postive numbers,and such $a+b+c+d=4$, show that ...
4
votes
0answers
81 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
4
votes
0answers
146 views

Prove that:$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geqslant a^n+b^n+c^n$

Find $n\in\mathbb{N}^+$ For all Positive real numbers $a,b,c$ sastifying $a+b+c=3$ $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geqslant a^n+b^n+c^n$
4
votes
0answers
212 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 ...
4
votes
0answers
741 views

Proving the Power Mean Inequality using Chebyshev's sum inequality

Question: Can Chebyshev's sum inequality be used to prove the generalized mean inequality, or at least a portion of it? If we let $$P(r) = \sqrt[r]{\frac{x_1^r + x_2^r + \cdots +x_n^r}{n}}$$, we have ...