Questions on proving, manipulating and applying inequalities.

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22
votes
0answers
465 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 ...
14
votes
0answers
282 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 ...
10
votes
0answers
120 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 $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
10
votes
0answers
949 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 ...
9
votes
0answers
208 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
830 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
189 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 ...
9
votes
0answers
215 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
194 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
181 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
266 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 ...
9
votes
0answers
161 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
102 views

An inequality for a trigonometric sum

Using the equirepartion of the sequence $(n \mod 2\pi)$ one can show that $$\lim_{n\to\infty} \frac1n \sum_{k=0}^n|\cos k|=\frac{2}{\pi}$$ Numerical evidence shows that, for every $n$, $$ ...
7
votes
0answers
146 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 ...
7
votes
0answers
621 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
228 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
139 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
6
votes
0answers
119 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
229 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
270 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
132 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 ...
5
votes
0answers
84 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 ...
5
votes
0answers
187 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 ...
5
votes
0answers
89 views

Does this inequality hold? Proof / Counterexample

Does the following inequality $ \int_0 ^\infty x^2 |\frac{d}{dx}f(x)|^2 dx - \int_0 ^\infty x |f(x)|^2 dx + 2\pi (\int_0 ^\infty x^2 |f(x)|^2 dx) (\int_0 ^\infty x |f(x)|^2 dx) > -\frac{1}{8\pi} ...
5
votes
0answers
119 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
124 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
266 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 ...
5
votes
0answers
334 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
107 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
161 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
205 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 ...
5
votes
0answers
155 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
177 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
74 views
+50

Proof of an inequality in $\mathbb{C}$

Let $z\in \mathbb{C}, n \geq 2$. Show this complex inequality $$|z^n-1|^2\le |z-1|^2\left(1+|z|^2+\dfrac{2}{n-1}\Re{(z)}\right)^{n-1}$$ For $n=2$ the inequality is easy to prove: $$|z^2-1|^2\le ...
4
votes
0answers
97 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 ...
4
votes
0answers
83 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
63 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 ...
4
votes
0answers
133 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
60 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
85 views

Minimal conditions to show $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$ $$G = ...
4
votes
0answers
118 views

ax+by+cz<=d, find the maximum value of ax+by+cz.

I am recently calculating that kind of questions as shown in title. And of course, I use a lot of time to find the answer. Therefore I would like to know how I can calculate faster. Q.: ax + by + ...
4
votes
0answers
177 views

Proof that $\oint_r d(x,N + n) < 0 $?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
4
votes
0answers
80 views

Prove this inequality $a^4+b^4+c^4+9\ge 4(a^2b+b^2c+c^2a)$

let $a,b,c>0$ and such $abc=1$, show that $$a^4+b^4+c^4+9\ge 4(a^2b+b^2c+c^2a)$$ Schur inequality $$a^4+b^4+c^4\ge \sum_{cyc}ab(a^2+b^2)-abc(a+b+c)$$ It suffices to show that ...
4
votes
0answers
66 views

Upper bounding a definite integral

So I have the following problem. Let $F$ be the set of functions for which $|f(x)| \le 2$ for all $x$ and $\int_{0}^{5} [f(x)]^2dx \le 16$. Over all the functions in $F$, compute the maximum ...
4
votes
0answers
84 views

About two combinatorial counting problems.

Here are the problems: Suppose $X$ is a set of $n$ elements, and $S_1,...,S_m$ are $m$ subsets of $X$ of average size at least $n/w$. Show that if $m\geq 2kw^k$, then there are $k$ distinct ...
4
votes
0answers
234 views

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

Consider the $M \times M$ complex positive semidefinite matrices ${\bf A}, {\bf B}, {\bf Z}$. We have the relation $\mu_{\text{max}}{\bf I} \succeq {\bf A} \succeq {\bf B} \succ \mu_{\text{min}}{\bf ...
4
votes
0answers
128 views

By which way I can prove $\arctan\left(2^{\sqrt{3}-\sqrt{7}}\right)<\frac{61}{125}$

By which way I can prove $$\arctan\left(2^{\sqrt{3}-\sqrt{7}}\right)<\frac{61}{125}$$
4
votes
0answers
80 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...