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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

22
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328 views

Prove that $\frac{1}{\phi}<\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx< \frac{24+\sqrt{2}}{41} $

I'm sure that's a coincidence, but the Laplace transform of $1/\Gamma(x)$ at $s=1$ turns out to be pretty close to the inverse of the Golden ratio: $$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0....
18
votes
0answers
486 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 ...
17
votes
0answers
469 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
15
votes
0answers
305 views

How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number ...
13
votes
0answers
532 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
13
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0answers
390 views

A very general method for solving inequalities repaired

Yesterday, I asked a question about a very general method for solving equations I had found here. As it turned out, there were quite some problems with my method and I got a lot of good feedback. ...
12
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0answers
430 views

How to determine the minimal constant $\lambda = \lambda(n,k)$

Given fixed positive integers $n,k$, determine the minimal constant $\lambda = \lambda(n,k)$ for which the following inequality holds for any $a_1,a_2,...,a_n>0$ (taking indices mod $n$ if required)...
12
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0answers
414 views

If $a+b+c = 3abc$ and $\frac17 \leq k \leq 7$ prove $ \frac1{ka+b}+\frac1{kb+c}+\frac1{kc+a} \leq \frac3{k+1} $

@Michael Rozenberg, in If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$, asks for a proof of one special case ($k=7$) of what I believe is a more general set of identities: ...
12
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265 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 ...
11
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0answers
647 views

If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$

Let $a$, $b$ and $c$ be positive numbers such that $a+b+c=abc$. Prove that: $$\frac{1}{7a+b}+\frac{1}{7b+c}+\frac{1}{7c+a}\leq\frac{\sqrt3}{8}$$ I tried C-S: $$\left(\sum_{cyc}\frac{1}{7a+b}\right)^2\...
11
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0answers
284 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 ...
11
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0answers
883 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
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0answers
271 views

Number of simultaneously solvable linear equations with nonnegative variables

Let $N$ variables $x_i \ge 0$ but not all of them be zero. One may fix $\sum_{i=1}^N x_i = 1$. There are $P$ equations which need to be solved, with coefficients $a^k_i$ indexed with superscripts $k =...
8
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382 views

Implications of inequalities

For $i=1,2,3$, consider a random variable $Y_i$ taking value in $$ \mathcal{Y}:=\{(1,1), (1,0), (0,1), (0,0)\} $$ and a random closed set $S_i$ taking value in $\mathcal{S}$ that is the power set of $...
8
votes
0answers
327 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 $\pi-...
8
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356 views

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

Question: let $a_{1},a_{2},\cdots,a_{n}>0$,and such $$a_{1}a_{2}\cdot \cdots a_{n}=1$$ prove or disprove: $$\dfrac{1}{a^5_{1}(a_{2}+2a_{3})^2}+\dfrac{1}{a^5_{2}(a_{3}+2a_{4})^2}+\cdots+\dfrac{...
8
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248 views

Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ \frac{(a+b)!^2}...
7
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264 views

I conjecture $3(a^ab^bc^c)^{\frac{1}{a+b+c}}\ge (a^ab^b)^{\frac{1}{a+b}}+(b^bc^c)^{\frac{1}{b+c}}+(c^ca^a)^{\frac{1}{c+a}}$

I Conjecture $$3\left(a^ab^bc^c\right)^{\dfrac{1}{a+b+c}}\ge \left(a^ab^b\right)^{\frac{1}{a+b}}+\left(b^bc^c\right)^{\frac{1}{b+c}}+\left(c^ca^a\right)^{\frac{1}{c+a}}$$ This conjecture is based on ...
7
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0answers
135 views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
7
votes
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387 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 ...
7
votes
0answers
339 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}+\...
7
votes
0answers
285 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 ...
7
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0answers
224 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}...
6
votes
0answers
45 views

Double inequality with a certain number of reals

I've encountered the following problem that I don't know how to solve: Given positive natural $n$ and positive real $x_1, x_2, ..., x_n$ prove that there exists such positive natural $N$ that $(1+\...
6
votes
0answers
150 views

Heisenberg's inequality

The Heisenberg's inequality in $\Bbb{R}$ reads $$\|f\|_{L^2}^4\leq \int_{\Bbb{R}}x^2f(x)^2dx\int_{\Bbb{R}}\xi^2\hat{f}(\xi)^2d\xi$$ where by $\hat{f}$ we refer to the Fourier transform of $f$. The ...
6
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0answers
301 views

Can one use properties of polynomials in order to generalize the generalized Cauchy-Schwartz inequality?

Sorry about the edits guys, I forgot to add binomial coefficients, I hope I didn't cause any needless confusion. Edit(again): I've been thinking about this a bit and perhaps I should clarify the ...
6
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0answers
119 views

Is there a combinatorial proof to this inequality?

I verified that this inequality: $$ \sum_{i=0}^{k-1} \sum_{j=0}^{k+1} {3 k-3\choose i} {3 k+3\choose j} \geq \sum_{i=0}^{k} \sum_{j=0}^{k} {3 k\choose i} {3 k\choose j} $$ holds for all $k$ between 1 ...
6
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0answers
146 views

How to prove or falsify this inequality?

In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi $ $$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$...
6
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347 views

Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
6
votes
0answers
197 views

Difficult inequality with $\pi$

My question is about an inequality ,originally I wanted to prove this : If $a,b,c,d >0$, and $a+b+c+d=4$, prove that $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi.$$ from here My approach is to use this ...
6
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0answers
391 views

Prove that $\frac{a+b+c}{3}\geq\sqrt[53]{\frac{a^4+b^4+c^4}{3}}$

Let $a$, $b$ and $c$ be non-negative numbers such that $(a+b)(a+c)(b+c)=8$. Prove that: $$\frac{a+b+c}{3}\geq\sqrt[53]{\frac{a^4+b^4+c^4}{3}}$$ I think $uvw$ does not help here. My another similar ...
6
votes
0answers
179 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 ...
6
votes
0answers
86 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}...
6
votes
0answers
173 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(...
6
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0answers
1k 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 ...
6
votes
0answers
341 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 ...
6
votes
0answers
190 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
486 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} \...
6
votes
0answers
2k 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 ...
6
votes
0answers
271 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{...
6
votes
0answers
329 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 ...
6
votes
0answers
206 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{...
5
votes
0answers
53 views

Are these upper and lower bounds for $\frac{x!}{\left\lfloor{x}\right\rfloor!}$ useful? If so, are they already known?

Truncating the infinite series for the derivative of the Digamma function $$ \psi'(x) = \sum_{n=0}^\infty\frac{1}{(x + n)^2} $$ after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ ...
5
votes
0answers
116 views

Proof of a technical fact in the book of Schapire and Freund on boosting

I am currently looking at Exercise 10.3, Chapter 10 of the book on Boosting by Schapire and Freund. More precisely, in the middle of the exercise they propose to use, without proof, the technical fact ...
5
votes
0answers
187 views

Proving that $q(A,S)>0$ if $A>0$, $S<0$ and $p_i(A,S)>0$

Consider the polynomials $$p_1=3(56-56A+A^2)-112(-6+A)S-14(-36+A)S^2+112S^3+7S^4,$$ $$p_2=-112(-6+A)-28(-36+A)S+336S^2+28S^3,$$ $$p_3=-28(-36+A)+672S+84S^2,$$ $$p_4=672+168S,$$ $$p_5=168,$$ $$q=-168S-...
5
votes
0answers
121 views

An annoying optimization problem

At first I thought the following problem looked simple, but I've had serious problems pinning it down: Suppose that $\prod_{i=1}^k x_i$ is fixed. Then find the minimum value of $$\sum_{i=1}^k (1 - ...
5
votes
0answers
80 views

Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base.

For $x \in \mathbb{R}^+$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $k \in \mathbb{N}$. Show that $2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$ for $k > 1$. ...
5
votes
0answers
141 views

Gronwall's inequality for higher order derivatives.

Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\...
5
votes
0answers
219 views

Von Neumann's Trace Inequality for Multiple Matrices

Von Neumann's trace inequality states that $|tr(AB)| \le \sum_{i=1}^{n} \sigma_i(A) \sigma_i(B)$ where $A, B$ are general square matrices with singular values $(\sigma_i(A)), (\sigma_i(B))$, ...
5
votes
0answers
94 views

An inequality with a constraint

Let $x_1,...,x_n , y_1,...,y_m$ be real numbers such that $\sum_{i=1}^n x_i=\sum_{j=1}^my_j=0$ . Then how to show that for any real numbers $a_1,...,a_n $ and $b_1,...,b_m$ , $2\sum_{i=1}^n \sum_{j=1}...