Questions on proving, manipulating and applying inequalities.

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24
votes
0answers
507 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 $$\sum_{i=1}^{n}(a_{i}+b_{i}+c_{i}+d_{i})\sum_{i=1}^{n}\dfrac{a_{i}b_{i}+b_{i}c_{i}+c_{i}d_{i}+d_{i}a_{i}+a_{i}c_{i}+b_{i}...
16
votes
0answers
542 views

Stronger than Nesbitt inequality

For $x,y,z >0$, prove that $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \sqrt{\frac94+\frac32 \cdot \frac{(y-z)^2}{xy+yz+zx}}$$ Observation: This inequality is stronger than the ...
15
votes
0answers
294 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 ...
11
votes
0answers
995 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 ...
10
votes
0answers
125 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 $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
9
votes
0answers
235 views

Interpretation of $\frac{22}{7}-\pi$

Integral and series proofs that $\frac{22}{7}>\pi$ We can prove that $\frac{22}{7}$ exceeds $\pi$ by using Dalzell integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ or its ...
9
votes
0answers
833 views

How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
9
votes
0answers
192 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation $$n!...
9
votes
0answers
227 views

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 ...
9
votes
0answers
202 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? ...
9
votes
0answers
185 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 $\...
9
votes
0answers
162 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 ...
8
votes
0answers
189 views

How prove this inequality $(\sum a_{1}^{1.5})^2\ge \sum a_{1}\sum a_{1}a_{2}$

Now my question let $a_{1},a_{2},\cdots,a_{n}$ are positive numbers,and $a_{n+i}=a_{i},i=1,2,\cdots$,show that $$(\sum a_{1}^{1.5})^2\ge \sum a_{1}\sum a_{1}a_{2}$$ my teacher (tian275461) have ...
7
votes
0answers
84 views

Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$

If $a,b,c,d >0$, and $a+b+c+d=4$, prove that $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi.$$ I don't think Jensen's inequality will help here, but I think first determining where equality holds will ...
7
votes
0answers
85 views

Use a direct proof to show that if $x\gt 1$ then $x^5 +x+1\gt 2$.

Use a direct proof to show that if $x\gt 1$ then $x^5 +x+1\gt 2$. It's obvious that if $x\gt 1$ then $x+1\gt 2$ , also $x^5\gt0 \;as \;x\gt1\gt0$, thus $x^5+x+1\gt 2$. Is it fine ? I can't ...
7
votes
0answers
219 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 ...
7
votes
0answers
185 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 k}...
7
votes
0answers
649 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
182 views

$a,b,c >0$, prove $\sqrt[2]{\frac{a}{b+c}}+\sqrt[3]{\frac{b}{c+a}}+\sqrt[4]{\frac{c}{a+b}} \geqslant \frac{7}{12} \cdot2^{\frac67} \cdot 3^{\frac47}$

$a,b,c >0$, prove $$\sqrt[2]{\frac{a}{b+c}}+\sqrt[3]{\frac{b}{c+a}}+\sqrt[4]{\frac{c}{a+b}} \geqslant \frac{7}{12} \cdot2^{\frac67} \cdot 3^{\frac47}$$ What I tried: 1) It seems like a Nesbitt ...
6
votes
0answers
230 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for $\...
6
votes
0answers
120 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
175 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
232 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
273 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
32 views

$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...
5
votes
0answers
73 views

Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
5
votes
0answers
126 views

How to prove that $a^2b+b^2c+c^2a \leqslant 3$, where $a,b,c >0$, and $a^ab^bc^c=1$

$a,b,c >0$, and $a^ab^bc^c=1$, prove $$a^2b+b^2c+c^2a \leqslant 3$$ I don't even know what to do with the condition $a^ab^bc^c=1$. At first I think $x^x>1$, but I was wrong. This inequality is ...
5
votes
0answers
145 views

Strict inequality for logarithmic integrals

Let $0<a<b<1$. Does the following inequality hold: $$\max_{f\in L^2[0,a],\,\,\|f\|_2=1}\Bigg|\int_0^a\int_0^af(x)f(y)\ln|x-y|dxdy\Bigg|$$ $$<\max_{g\in L^2[0,b],\,\,\|g\|_2=1}\Bigg|\...
5
votes
0answers
121 views

Prove with some AM-GM inequality?

I have proved the following inequality: Let $a,b,c>0$ $$\dfrac{(a+\sqrt{ab}+\sqrt[3]{abc})}{3}\le \sqrt[3]{a\cdot\dfrac{a+b}{2}\cdot\dfrac{a+b+c}{3}}$$ My solution is:$$a\dfrac{a+b}{2}\cdot\dfrac{...
5
votes
0answers
92 views

An integral to prove that $\log(2n+1) \ge H_n$

Dalzell integral The equation $$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ proves that $\frac{22}{7}-\pi>0$ because the integrand is positive. Some Dalzell-type integrals for $\log(...
5
votes
0answers
191 views

upper bounding alternating binomial sums

So we know that $\large\sum_\limits{i=0}^t\dbinom{m}{i}\dbinom{n-m}{t-i}=\dbinom{n}{t}$ by a simple counting argument. Now is there any bound on the quantity $\large\sum_\limits{i=0}^t(-1)^i\dbinom{...
5
votes
0answers
123 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 $L^1$ ...
5
votes
0answers
412 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 ...
5
votes
0answers
130 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
270 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 $$\dfrac{a}{\sqrt{a+3b}}+\dfrac{b}{\sqrt{b+3c}}+\dfrac{c}{\sqrt{c+3d}}+\dfrac{d}{\sqrt{d+3a}}\le\dfrac{a^2}{a+b^2}+\dfrac{b^2}{b+c^2}+\...
5
votes
0answers
340 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
108 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
162 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
210 views

Solving a system of linear inequalities

I have a personal problem I want to solve. I have a system of linear inequalities with 97 unknowns and 150000 ineqaulities, I think formal notation of the problem should be something like this \begin{...
5
votes
0answers
157 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 $f(...
5
votes
0answers
178 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)} + \frac{...
4
votes
0answers
90 views

Prove $\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$

$a,b,c \geqslant 0,$$ a+b+c=3$, and $(a+b)(b+c)(c+a) \neq 0$ , prove $$\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$$ I try Bernouli's ...
4
votes
0answers
90 views

A Matrix Norm Inequality $\|A^{1/2}B^{1/2}(A+B)^{-1/2}\|_F \geq \|A^{1/2}(A+B)^{-1/2}B^{1/2}\|_F$

Let $\|X\|_F:= \sqrt{\text{Tr} \left(XX^\dagger\right) }$ denote the Frobenius norm. Does anyone know how to show the norm inequality: $\left\|A^{\frac12}B^{\frac12}(A+B)^{-\frac12}\right\|_F \geq ...
4
votes
0answers
97 views

$a^{|b-a|}+b^{|c-b|}+c^{|a-c|} > \frac52$ for $a,b,c >0$ and $a+b+c=3$

$a,b,c >0$ and $a+b+c=3$, prove that $$a^{|b-a|}+b^{|c-b|}+c^{|a-c|} > \frac52$$ What I did: It is cyclic inequality so I assume $c= min\{ a,b,c \}$. I consider the first case where $a>b&...
4
votes
0answers
106 views

Prove $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$

$a,b,c >0$ and $a+b+c=3$, prove $$\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$$ I try to apply AM-GM $$\left(...
4
votes
0answers
149 views

(2016 China team selection Test) with a complex inequality

Let $z_{1},z_{2},z_{3}$ be complex numbers, such that: $z_{1}+z_{2}+z_{3}=0,|z_{i}|<1,i=1,2,3$. Find the minimum of the positive $A$ such that: $$|z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}|^2+|z_{1}z_{2}z_{...
4
votes
0answers
84 views

Inequalities involving polynomials with combinatorial coefficients

For all non-negative integers $i$ and $j$ such that $j\leq i$, define the array of polynomials $$p_{ij}(z):=\sum_{h=(j-1)_+}^{i-1} {i\choose h}{i-j\choose{i-h-1}}z^h,$$ where $(a)_+=\max\{a,0\}$ (we ...
4
votes
0answers
138 views

How to prove this inequality (already verified by numerical simulation)?

I have a conjecture which has been verified extensively by simulation. The conjecture is as follows: $\forall t \in [0, 1], \alpha \in [0,1]$, and positive real sequences $\{p\}_{i:1,\dots,n}, $, $\{...
4
votes
0answers
61 views

Proof of inequality involving binomial coefficients

Proving things is not my forte. Stumbled upon following identity: $$\binom{n+k+1}{k}>\sum_{i=1}^k \binom{\alpha_i}{i}$$ For $\alpha_i<\alpha_j $ for $i<j$ $0\leq \alpha_i \leq n+k$ Also $...
4
votes
0answers
91 views

Finding the minimum value of a radical expression

If $a$, $b$ and $c$ are positive real numbers, find the minimum value of $\sqrt { \frac { a }{ b+c } } +\sqrt [ 3 ]{ \frac { b }{ c+a } } +\sqrt [ 4 ]{ \frac { c }{ a+b } } $. I am not able to ...