Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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34
votes
1answer
504 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....
20
votes
0answers
529 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 ...
18
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0answers
522 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 ...
15
votes
1answer
253 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 $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
14
votes
1answer
563 views

$\sum\limits_i a_i^2\sum\limits_i b_i^2+\left(\sum\limits_ia_i b_i\right)^2\geq \sqrt{\sum\limits_i a_i^4\sum\limits_i b_i^4}+\sum\limits_ia_i^2b_i^2$

I have no idea about how to prove (or disprove) the following inequality: $$ \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)+\left(\sum_{i=1}^na_i b_i\right)^2\geq \sqrt{\left(\sum_{i=1}...
13
votes
0answers
575 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
votes
1answer
202 views

Is there any simple ways to compare $x^y$ and $y^x$ without a calculator?

There are plenty of discussion on MSE about how to compare $x^y$ and $y^x$. For $x,y>e$, it is sufficient to just compare $x$ and $y$ to reach a conclusion. But I wonder if there are some general ...
13
votes
0answers
451 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
votes
0answers
451 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
votes
1answer
320 views

How to prove this integral inequality $ \int_0^{2\pi} p(x)[p(x)+p''(x)] dx \int_0^{2\pi}\frac{1}{p(x)+p''(x)} dx\geq 2\pi \int _0^{2\pi} p(x) dx $?

Let $p\in C^2(\mathbb{R})$ be a $2\pi$-periodic function such that $p(x)>0$ and $p(x)+p''(x)>0$ for all $x\in \mathbb{R}$. Then it holds $$ \int_0^{2\pi} p(x)[p(x)+p''(x)] dx \int_0^{2\pi}\frac{...
12
votes
0answers
907 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. ...
12
votes
1answer
238 views

Inequality with summation of cosine terms $\left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| \leq 1 + 2\sum_{j=1}^k \cos (\frac{2\pi }{q}j)$

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| ...
12
votes
1answer
2k 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 ...
12
votes
0answers
275 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 ...
12
votes
1answer
451 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 ...
12
votes
2answers
412 views

show that $\sum_{k=1}^{n}(1-a_{k})<\frac{2}{3}$

Let $a_{1}=\dfrac{1}{2}$, and such $a_{n+1}=a_{n}-a_{n}\ln{a_{n}}$,show that $$\sum_{k=1}^{n}(1-a_{k})<\dfrac{2}{3}$$ My attemp: let $1-a_{n}=b_{n}$,then we have $$b_{n+1}=b_{n}+(1-b_{n})\ln{(1-b_{...
11
votes
0answers
564 views

Refinement of a famous inequality

I refine a famous inequality this is the following : Let $x,y>0$ then we have : $$x^n+y^n\leq \Big(\frac{x^n+y^n}{x^{n-1}+y^{n-1}} \Big)^n+\Big(\frac{x+y}{2}\Big)^n$$ It's equivalent to : ...
11
votes
1answer
101 views

Reference Request for inequalities without variables

Prove that $\dfrac{1}{2} < \dfrac{1}{101} + \dfrac{1}{102} + \dots + \dfrac {1}{200} < 1$ I started a math club at my school in the hopes of promoting math interest, and I want to interest my ...
11
votes
0answers
315 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 ...
10
votes
1answer
261 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 ...
9
votes
0answers
249 views

(Dis)Prove $\sum_{i=1}^n\sum_{j=1}^n{(|x_{i}-x_{j}|-|y_{i}-y_{j}|)^2}\geq 4$

Let $n\ge 4$ and two vectors $x$ and $y$ in $\mathbb{R}^n$ that satisfy $\sum_{i=1}^{n}{x_{i}^2}=\sum_{i=1}^{n}{y_i}^2=1$ $\sum_{i=1}^{n}{x_{i} y_i}=0$ $\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$ ...
9
votes
0answers
158 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 $...
9
votes
1answer
324 views

Any similar Lagrange's identity inequality

Following problem I have post MO ,we know Lagrange's identity $$(a^2_{1}+a^2_{2}+a^2_{3})(b^2_{1}+b^2_{2}+b^2_{3})=(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})^2+\sum_{i=1}^{2}\sum_{j=i+1}^{3}(a_{i}b_{j}-a_{...
9
votes
1answer
392 views

$a,b,c >0$, and $ab+bc+ca=3$, prove $(a^ab^bc^c)^{\frac{3}{a+b+c}} \geqslant \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$

$a,b,c >0$, and $ab+bc+ca=3$, prove $$(a^ab^bc^c)^{\frac{3}{a+b+c}} \geqslant \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$$ I think the equality is only achieve when $a=b=c=1$. The condition $ab+bc+ca=3$ is ...
9
votes
0answers
257 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}...
8
votes
1answer
83 views

Proving $\sin(\tanh x) \ge \tanh(\sin x)$, for $x \in [0,\pi/2]$

Earlier, a very interesting proof of an inequality has been proposed at MSE: How prove this inequality $\tan{(\sin{x})}>\sin{(\tan{x})}$ Here the question is: How to prove that $$\sin(\tanh x) \ge ...
8
votes
0answers
195 views

Is it true that $\phi(\mu)=\mu F(\mu)^2-\int_{\mu}^{\overline{v}}F(v)[1-F(v)]dv\geq 0$?

Consider a random variable $V$ with distribution function $F$ and density function $f$ with support $[\underline{v},\overline{v}]$, where $0\leq\underline{v}<\overline{v}$. The mean is $\mu$. Here $...
8
votes
0answers
287 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
votes
0answers
182 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 ...
8
votes
0answers
388 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
280 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 ...
8
votes
1answer
138 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 $$(a+...
8
votes
0answers
452 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
votes
0answers
237 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}...
8
votes
1answer
121 views

Given $\sum\limits_{i=1}^6a_i^2=6$, where $a_i>0$, $a_7=a_1$. Prove that $\sum\limits_{i=1}^6\frac{a_i^2}{a_{i+1}}\geq6$

Let $a_i$ be positive numbers such that $a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2=6$. Prove that: $$\frac{a_1^2}{a_2}+\frac{a_2^2}{a_3}+\frac{a_3^2}{a_4}+\frac{a_4^2}{a_5}+\frac{a_5^2}{a_6}+\frac{a_6^2}{...
8
votes
1answer
420 views

Using Jensen's inequality to prove another inequality?

Suppose $u(\cdot)$ and $v(\cdot)$ are two differentiable, strictly increasing, and strictly concave real functions. Specifically, $v(\cdot)$ is "more concave" than $u(\cdot)$ in the sense that there ...
7
votes
0answers
212 views

show that $\left(\sum_{k=1}^{n}x_{k}\tan{k}\right)\left(\sum_{k=1}^{n}x_{k}\cot{k}\right)\le n^3\sum_{k=1}^{n}x^2_{k}$

let $x_{k}>0(k=1,2,3,\cdots,n)$ show that $$\left(\sum_{k=1}^{n}x_{k}\tan{k}\right)\left(\sum_{k=1}^{n}x_{k}\cot{k}\right)\le n^3\sum_{k=1}^{n}x^2_{k}$$ My try use AM-GM and Cauchy-Schwarz ...
7
votes
1answer
139 views

Find the maximum of the $f=\sum_{i=1}^{n}\left(\binom{a_{i}}{2}\cdot\sum_{j<k,j,k\neq i}a_{j}a_{k}\right)$

Let $a_{1},a_{2},\ldots,a_{n}$ be integers, such that $a_{i}\ge 0$ for $i=1,2,\cdots,n$, and such that $\sum_{i=1}^{n}a_{i}=120$. Find the maximum value of $$F=\sum_{i=1}^{n}\left(\binom{a_{i}}{2}\...
7
votes
0answers
343 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-...
7
votes
1answer
194 views

Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$

Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Prove that $$S = x \sin^2 A + y \sin^2 B + z \sin^2 C \leq \dfrac{\left(yz+zx+xy\right)^2}{4xyz}$$ and find ...
7
votes
1answer
155 views

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

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{...
7
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 ...
7
votes
0answers
298 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
1answer
242 views

Disjoint sets with twice ratio

Given are a positive integer $n$ and positive real numbers $a_1,\dots,a_n,b_1,\dots,b_n$. A subset $S\subseteq N=\{1,\dots,n\}$ is called $a$-good if $$\sum_{i\in S}a_i\geq \frac{1}{2}\left(\sum_{i\in ...
6
votes
0answers
50 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
121 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 ...
6
votes
1answer
151 views

Showing $\int_0^s \int_0^u (F_1(u)-F_1(u-v)) f_1(u-v) e^{-v}\, dv\, du \leq \int_0^s \int_0^u (F_2(u)-F_2(u-v)) f_2(u-v) e^{-v}\, dv\, du$

Question Let $F(s)$ be a cumulative distribution function (cdf) of a random variable on $[0,\infty)$ which only has an atom at $0$, i.e. $F(0) >0$ and for all $s>0$:$$ \lim_{h\rightarrow 0}F(s+...
6
votes
0answers
211 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
votes
1answer
80 views

Generalization of convexity

This is not a homework, this is just something that came to my mind recently. Assume $f$ is a sufficiently nice function. We know that $$\frac{df}{dx} \geq 0 \iff f(x_2) \geq f(x_1) \text{ for } x_2 \...
6
votes
0answers
356 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 ...

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