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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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Ways of finding upper bounds of hypergeometric functions

I realized that I don't really know any good ways of finding bounds of hypergeometric functions after ${}_2 F_1$. For example, numerical evaluation convinced me that the generalized hypergeometric ...
Loading - 146 Complete's user avatar
4 votes
0 answers
25 views

Is there a Log-Sobolev inequality for Lebesgue measure on $[0,1]$ on compact subsets of $\mathbb R^n$?

After searching on the internet for long enough, I would like to pose the question here. I hope there is no duplicate (if there is please let me know) Is it true that, there is a universal constant $C&...
mathnoob's user avatar
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-1 votes
1 answer
51 views

Maximum value of $a_{1}a_{2}+a_{2}a_{3}+\cdots +a_{7}{a_{8}}$

If $a_{1},a_{2},\cdots \cdots ,a_{8}$ all are non negative numbers and $a_{1}+a_{2}+\cdots +a_{8}=16$. Then maximum of $\displaystyle a_{1}a_{2}+a_{2}a_{3}+\cdots +a_{7}a_{8}$ What I try : $\...
jacky's user avatar
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-1 votes
1 answer
33 views

Proving opposite statements in inequalities

I was solving the following inequality: "Considering $a,b,c>0$ satisfying $abc=1$, prove that: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge a+b+c$$ I tried to use Hölder: $$\left(\frac{a}{b}+\frac{...
Francisco Sierra's user avatar
2 votes
1 answer
94 views

Show that for all real numbers $x_1, x_2, \ldots, x_n$, $\sum_{i=1}^n \sum_{j=1}^n\left|x_i+x_j\right| \geq n \sum_{i=1}^n\left|x_i\right| $ [duplicate]

Show that for all real numbers $x_1, x_2, \ldots, x_n$, $$ \sum_{i=1}^n \sum_{j=1}^n\left|x_i+x_j\right| \geq n \sum_{i=1}^n\left|x_i\right| $$ If we fix $x_2,.....x_n$, and vary $x_1$, then we can ...
math_learner's user avatar
3 votes
1 answer
108 views

Prove $\sqrt{\frac{a+bc}{b+c}}+\sqrt{\frac{b+ac}{a+c}}+\sqrt{\frac{c+ba}{b+a}}\ge 3.$

Let $a,b,c$ be nonnegative real numbers such that $a+b+c+abc=4$, prove that $$\sqrt{\frac{a+bc}{b+c}}+\sqrt{\frac{b+ac}{a+c}}+\sqrt{\frac{c+ba}{b+a}}\ge 3.$$ I've seen it here (#10). I tried Holder ...
30 Anh Ti 711's user avatar
1 vote
1 answer
45 views

A variant of Shapiro inequality

Let $n$ be a natural number and $x_1,x_2,\dots,x_n$ be positive reals. Let $x_0=x_n$ and $x_{n+1}=x_1$. Is it always true that $$\sum_{i=1}^n\frac{x_i}{x_{i-1}+x_{i+1}}\ge\dfrac n2$$ I know when $n\...
Lucenaposition's user avatar
-2 votes
0 answers
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Recurring A.M-G.M. in limiting case [closed]

I came across a question where a recurring AM-GM sequence converges in the limit, but I am not able to find point of convergence as a closed form expression. Question: Let $f(x)$, $g(x)$ be defined as:...
Shivang Gupta's user avatar
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0 answers
23 views

One Confusion in Proving Brezis-Lieb Lemma

Here is the Brezis-Lieb Lemma: $(X,\mathfrak{A},\mu)$ is a measure space, consider $L^{p}(X):p \in(0,\infty)$ . Then $\left\{ f_{n} \right\}$ is a sequence of extended complex-values measurable ...
M_k's user avatar
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Proving a the distance between Cauchy sequences converges [duplicate]

Assume we have two Cauchy sequences { $x_n$ } and {$y_n$} in the metric space $(X,d)$. Is it true that the sequence {$a_n$}$=d(x_n,y_n)$ is convergent in $\mathbb{R}$? Here is my try: $$$$ Since those ...
Krum Kutsarov's user avatar
-4 votes
1 answer
51 views

A rational function dominated by a linear function on a certain interval [closed]

Considering $\alpha, \beta, \gamma, c > 0$, $\alpha - c > 0$, and $\mu_0 = \frac{\gamma (\alpha - c + \gamma)}{4 \beta}$, we have the following two functions: $$ f(\mu) = \frac{(\alpha - c + \...
Serkewtin's user avatar
0 votes
0 answers
45 views

Given that $x_i\ge0$ and $0<p<1$, find an upper bound for $\sum_{i=1}^n x_i^p$ as a function of $a:=\sum x_i$ [duplicate]

Suppose $\boldsymbol{x}$ is an $n$-dimensional vector and all elements are nonnegative with the condition that $\sum_{i=1}^nx_i= a$. I am wondering how it is possible to find an upper bound on $\sum_{...
Amin's user avatar
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2 votes
2 answers
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If $x > 0$ is a real number and $x+\frac{1}{x} \leq 7$ then the maximum value of the expression $x^2-\frac{1}{x^2}$ is...?

The problem If $x > 0$ is a real number and $x+\frac{1}{x} \leq 7$ then the maximum value of the expression $x^2-\frac{1}{x^2}$ is...? my idea We know that $x^2-\frac{1}{x^2}= (x- \frac{1}{x})(x+ \...
IONELA BUCIU's user avatar
6 votes
1 answer
357 views

"Peeling Technique" in Probability

So I am reading "Bandit Algorithms" by Lattimore wherein for one of the proofs he uses a technique called as "Peeling Device" which he says is a widely used tool in probability. I ...
tango's user avatar
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1 vote
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Simple proof (avoiding Gamma function) for a Gaussian lower bound?

Let $g \sim N(0, I_n)$ be a standard multivariate Gaussian vector in $\mathbb{R}^n$. It can be shown via use of Gamma function identities and inequalities that $$ \sqrt{\frac{n}{n+1}} \leq \mathbb{E}\...
Drew Brady's user avatar
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0 votes
1 answer
37 views

An inequality for a bisected "shifted quadrant" under a continuous symmetric bivariate distribution?

Suppose a bivariate probability distribution is continuous and has circular symmetry about the origin; i.e., the lines of constant density are concentric circles centered on the origin. Now consider ...
r.e.s.'s user avatar
  • 15.1k
2 votes
1 answer
147 views

Inequalities and averages

Dikshant writes down $2 k+1$ positive integers in a list where $k$ is a positive integer. The integers are not necessarily all distinct, but there are at least three distinct integers in the list. The ...
aiman's user avatar
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-1 votes
2 answers
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Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$

The problem Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$ My idea $(\sqrt{a}+\sqrt{b})^2=a+b+ 2\cdot \sqrt{ab}= 101 + 2\cdot \sqrt{ab}$ ...
IONELA BUCIU's user avatar
1 vote
1 answer
95 views

How to prove this simple inequality based on convexity of $e^{x}$?

Suppose $\theta > 0$ and $x>0$. I would like to show that $$ e^{\theta(x+1)} - e^{\theta x} - \frac{ e^{\theta x}-1}{x} \geq \frac{e^{\theta x}-1}{x} - (1-e^{-\theta}) $$ Another way to put it: ...
unknowngoogle's user avatar
0 votes
1 answer
23 views

fractional power function inequality [closed]

I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result. Let ...
Jabar S. Hassan's user avatar
0 votes
1 answer
75 views

Differential inequation $f'(x) > f(x)/x$

I am interested in the following differential inequation, for $f : \mathbb{R} \to \mathbb{R}$: $$f'(x) > \dfrac{f(x)}{x}$$ I only care about this being satisfied at infinity, ie for $x \gg 1$. ...
Azur's user avatar
  • 2,311
0 votes
4 answers
160 views

Show that: $(\frac{MN}{MA})^2+ (\frac{MP}{MB})^2+ (\frac{MQ}{MC})^2+ (\frac{MR}{MD})^2 \geq \frac{4 }{9}$

The problem Let $M$ be a point inside the tetrahedron $ABCD$. We denote by $N,P,Q,R$ the intersections of the lines $AM,BM,MC,DM$ with the planes $(BCD),(ACD),(ABD)$, respectively $(ABC)$. Show that: $...
IONELA BUCIU's user avatar
0 votes
0 answers
39 views

Same number of lists of integers [duplicate]

Have a following problem for which I'll show my reasoning (the problem is $1.6$ from book Problem solving methods in combinatorics by Pablo Soberon): If we want to write all the lists of length $n$ ...
slomil's user avatar
  • 176
0 votes
2 answers
118 views

better method of solving quadratic / cubic

Problem : Let $$\begin{align} f(x) &= x^4 - 8x^3 + 18x^2 \\ g(x) &= 9x^2 - 64x\end{align}$$ . Define $h : \mathbb{R}^+ \to \mathbb{R}$, $h(x)=f(x)-ag(x)$ for some real number $a$. If $h(x)$ ...
bFur4list's user avatar
  • 2,761
3 votes
0 answers
114 views

AM-GM inequality for non necessary positive numbers

For nonnegative real numbers $x_1,\cdots,x_n$ ($n\geqslant2$), it is well known that : $$n\prod_{i=1}^nx_i\leqslant\sum_{i=1}^nx_i^n\tag{$\star$}$$since this is equivalent to the AM-GM inequality. But ...
Adren's user avatar
  • 7,702
0 votes
1 answer
128 views

Show that $\frac{1-xy-x}{x+y+3} + \frac{1-zy-y}{z+y+3}+ \frac{1-xz-z}{x+z+3} \geq \frac{5}{11}$

The problem a) Show that $\frac{ab}{a+b}+ \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$, with equality for $ad=bc$ (solved already) b) Let the real numbers $x,y,z \in (0, \infty)$ with $x+y+z=1$. ...
IONELA BUCIU's user avatar
1 vote
1 answer
51 views

Finding a value $n$ such that $\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$ is true.

For what value of $n \in \mathbb{N}$ such that the following inequality is true. $$\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$$ Where $0<x\le \sqrt[5]{216}$ ATTEMPT: This is my first time tackling ...
JAB's user avatar
  • 323
0 votes
1 answer
35 views

Proof of Khintchine’s inequality for sub-gaussians

I am trying to prove the exercise 2.6.6 of HDP book by Roman Vershynin. The exercise is as follows. Actually the right hand side is easy to find, and the left hand side is also easy if given the red-...
dhliu's user avatar
  • 25
0 votes
0 answers
41 views

The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them

Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that $$ d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0} $$ Here we have the following definitions: For any (...
Saaqib Mahmood's user avatar
1 vote
0 answers
45 views

Estimation of Sum of Binomial Coefficients that doesn't Start from Zero

I am having some trouble showing the following estimation: Let $\mu \in (0, 1)$ be an absolute constant. There exists an absolute constant $\eta \in (0, 1)$ small enough such that the number of ...
Partial T's user avatar
  • 593
3 votes
0 answers
54 views

An integral inequality involving the Bernoulli polynomials

The classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{z\operatorname{e}^{t z}}{\operatorname{e}^z-1}=\sum_{j=0}^{\infty}B_j(t)\frac{z^j}{j!}, \quad |z|<2\pi. \end{...
qifeng618's user avatar
  • 1,846
0 votes
1 answer
60 views

A problem on dirt displacement

Definition. Given a function $f\in L^1(\mathbb{R})$ such that $xf\in L^1(\mathbb{R})$, the quantity $\int_\mathbb{R}xf(x)\,dx$ is called the unnormalized center of mass of $f$ and is denoted $UCM(f)$. ...
aleph2's user avatar
  • 984
3 votes
0 answers
91 views

A trigonometric maximum problem involving trigonometric constraints

Let $a,b,c,\alpha,\beta\in\mathbb{R}^+$ and $\alpha+\beta<2\pi$. Prove that if and only if $$\frac{\sin\alpha}{a\sqrt{b^2+c^2-2bc\cos\alpha}}=\frac {\sin\beta}{b\sqrt{a^2+c^2-2ac\cos\beta}}=-\frac{\...
Mr.He's user avatar
  • 579
4 votes
2 answers
342 views

Show that $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$

Let the real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=3$. Show that: $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$. My idea: First of all, I thought ...
IONELA BUCIU's user avatar
3 votes
1 answer
59 views

$a+b\sqrt{2}>1, a^2-2b^2=\pm 1 (a,b\in \mathbb{Z}) \implies a+b\sqrt {2}\geq 1+\sqrt {2}$

$\textbf{Example}:$ Let $K=\mathbb{Q}(\sqrt 2)$. We claim that $1+\sqrt 2$ is the fundamental unit of $K$. Easy to show that $N(1+\sqrt 2)=-1$ and thus a unit. Remain to show that if $v>1$ is any ...
Bowei Tang's user avatar
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1 vote
0 answers
62 views

Reference for $\sum_{m=1}^p\sum_{n=1}^q\frac2{\cos(2m\pi/p)+\cos(2n\pi/q)}\le pq(|p-q|+1)$ with coprime odd positive integers $p$ and $q$? [closed]

I am having trouble with the following problem that I found in this Art of Problem Solving post, and I would like some help to find a reference for it. Let $p$ and $q$ be coprime odd positive integers....
Curious's user avatar
  • 37
1 vote
0 answers
72 views

Prove that $\displaystyle \sum\limits_{i=1}^3\sqrt{ \sum\limits_{j=1}^3a_{ji}^2}\leq\sqrt{2}f(a_{11},\cdots,a_{33})$

For a $3\times3$ matrix $A=(a_{ij})$, let \begin{aligned}&f(a_{11},a_{21},a_{31},a_{12},a_{22},a_{32},a_{13},a_{23},a_{33})\\=&\text{max}\{|a_{11}+a_{21}+a_{31}|+|a_{12}+a_{22}+a_{32}|+|a_{13}+...
grj040803's user avatar
  • 701
2 votes
2 answers
77 views

Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

Show that for any positive real number $x$ the inequality holds: $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$ where by $[a], \{a\}$ we mean the whole par and fractional ...
IONELA BUCIU's user avatar
5 votes
1 answer
197 views

Finding the integer part of a sum

I want to find the integer part of $$\sum_{n=1}^{10^9}\frac{1}{n^{2/3}}=S$$ I know there is a way using integration but I tried using a different approach. I saw this approach with square roots but I ...
Vedant Lohan's user avatar
1 vote
0 answers
21 views

Upper bound for distribution function for variable with zero expectation. [duplicate]

A problem from final Year 1 probability exam. Is it true for any random variable $Y$ s.t. $E[Y]=0$ and $E[Y^2]<\infty$ that: $P(Y>x)\leq\frac{E[Y^2]}{E[Y^2]+x}$ ? I thought we can rewrite it ...
innerproduct's user avatar
1 vote
2 answers
76 views

Find min and max of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$

For $a,b,c \in \mathbb{R}$, $a^2 + b^2 + c^2 ⩽ 2$ Find min and max value of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$ I don't understand how to find min and max value of an absolute value sign. ...
trum fi fai's user avatar
0 votes
0 answers
17 views

Local property of an integration inequality to global result

Here is the question. Let $f$, $g$ be locally integrable functions on $\mathbb{R}^n$ such that $$\inf_{a \in \mathbb{R}} \int_{B} |f(x) - a|\ dx \leqslant \int_B|g(x)|\ dx$$ for all balls $B$ in $\...
ZYZ's user avatar
  • 1
6 votes
0 answers
169 views

Show that $(1-z_1\overline{w_1})(1-z_2\overline {w_2})+(1-z_1\overline {w_2}) (1-z_2\overline {w_1})$ is non-vanishing.

Show that $\left (1-z_1\overline{w_1} \right ) \left (1-z_2\overline {w_2} \right )+ \left (1-z_1\overline {w_2} \right ) \left (1-z_2\overline {w_1}\right )$ is non-vanishing (does not take the value ...
Anacardium's user avatar
  • 2,612
1 vote
1 answer
48 views

solution-verification | Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$

the problem a) Show that, for any real number $x$, $x^4-4x^3+4x^2+3>0$ b) Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$, where x is a number real my idea So ...
IONELA BUCIU's user avatar
-1 votes
4 answers
91 views

How to find the range of a quadratic function we can't use the quadratic formula?

In the question $f(x) = \frac {x}{(1+x^2)}$ $ yx^2 - x + y = 0 $ $x = \frac {1 ± \sqrt {1-4y^2}}{2y}$ We know that $x$ belongs to $\Bbb R$. So, $\frac {1 ± \sqrt {1-4y^2}}{2y}$ also belong to $\Bbb R$....
The's user avatar
  • 1
0 votes
0 answers
22 views

Why is the von Neumann inequality not always fulfilled for n-tuples of commuting contractions? Why cant we just take the single dilations and get it?

Let $(T_1, \ldots, T_n)$ be an $n$-tuple of contractions in a Hilbert space $H$. We know, that for every single $T_i$, there exists a unitary operator $U_i$ in a Hilbert space $K_i$, such that $$T_i = ...
S-F's user avatar
  • 41
1 vote
1 answer
54 views

Prove/disprove $\sum_{i=1}^n \left\lfloor {x_i}^{y} \right\rfloor \leq \left\lfloor x^y\right\rfloor$ for $x_i,y\geq 1.$

If $x_i,y\in\mathbb{R}_{\geq 1},\ \displaystyle\sum_{i=1}^n x_i = x,$ then it is obviously true that $\displaystyle \sum_{i=1}^n {x_i}^y \leq x^y,$ due to the Binomial theorem. After trying various ...
Adam Rubinson's user avatar
0 votes
0 answers
75 views

Show that $\frac{(a+1)(a-b)}{b+1}+ \frac{(b+1)(b-c)}{c+1} + \frac{(c+1)(c-a)}{a+1} \geq 0$

The problem Let $a,b$ be some real nonzero numbers. Show that: a) $a^2 \geq \frac{b(a+1)^2}{b+1}-b$ b) $\frac{(a+1)(a-b)}{b+1}+ \frac{(b+1)(b-c)}{c+1} + \frac{(c+1)(c-a)}{a+1} \geq 0$ my idea I was ...
IONELA BUCIU's user avatar
-2 votes
0 answers
30 views

System of nonlinear equations with squares and cubes [closed]

I am looking for solutions in $\mathbb{R}^+$ to the following system of equations: $a^2+b^2+c^2=2025$ and $\frac{a^3}{b}+c+\frac{b^3}{a}+c+\frac{c^3}{a}+b \leq\frac{1}{2}(a^2+b^2+c^2)$.
Ana's user avatar
  • 1
3 votes
1 answer
96 views

How to prove Bernstein's inequality?

I am currently learning the basics of machine learning and have come across Bernstein's inequality. This inequality is particularly useful in understanding the behavior of averages of random variables....
Jimmy Zhao's user avatar

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