Questions on proving, manipulating and applying inequalities.

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16 views

Bisections of the unit line inequality

Let $x_1 \in (0, 1)$. Iteratively define intervals $I_1,I_2,...$ and points $x_2,x_3,...$ by: $I_k$ is the longest sub-interval of $(0, 1)$ not containing any of the points $x_i , 1 \leq i \leq k$, (...
2
votes
5answers
73 views

Prove that $\frac{1}{x^{1+\epsilon}}<\frac{1}{x(\log x)^p}$

Given $p>0$, $\epsilon>0$, prove that $\displaystyle \frac{1}{x^{1+\epsilon}}<\frac{1}{x(\log x)^p}$ for sufficiently large $x$. If $p\leq \epsilon$, then $(\log x)^p\leq x^p\leq x^\epsilon$...
0
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0answers
32 views

When is the following trace inequality valid?

I have $A = A^T$ (and can have any real eigenvalue) and $B = B^T \succeq 0$ and want to know if the following holds $$ trace(AB) \leq 0 \iff \lambda_{max} (AB) \leq 0 $$ I know that the matrix $AB$ ...
1
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1answer
80 views

Proving triangle inequality using complete-linkage between clusters and arbitrary dissimilarity measure

Assuming a dissimilarity measure d satisfies the usual properties, I need to prove that complete linkage ( i.e. d(A,B)=maxx∈A,y∈B{d(x,y)} ) either satisfies or does not satisfy the triangle inequality ...
4
votes
1answer
48 views

What is the best time complexity of checking the inequality $a_1x_1 + \cdots + a_mx_m \le K$ to have a non-negative integer solution?

Consider $$a_1x_1 + \cdots + a_mx_m \le K$$ with $a_1, a_2, \ldots , a_m$ and $K$ being integers. I only need to know if the inequality has an integer solution or not. It means that there is ...
8
votes
2answers
703 views

Prove $\sum_{cyc}\left(\frac{a^4}{a^3+b^3}\right)^{\frac34} \geqslant \frac{a^{\frac34}+b^{\frac34}+c^{\frac34}}{2^{\frac34}}$

When $a,b,c > 0$, prove $$\left(\frac{a^4}{a^3+b^3}\right)^{\frac34}+\left(\frac{b^4}{b^3+c^3}\right)^{\frac34}+\left(\frac{c^4}{c^3+a^3}\right)^{\frac34} \geqslant \frac{a^{\frac34}+b^{\frac34}+c^{...
1
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0answers
19 views

Bounding a Sobolev norm on product manifold

Suppose you have a closed manifold $M$ and a function $f:M\times M\rightarrow\mathbb{C}$ that is $C^2$ in the $M$ variables, and suppose that at any point $q\in M$, we have $$f(\cdot,q)\in H_m(M).$$ ...
3
votes
5answers
317 views

Log concavity of binomial coefficients: $ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $

How do we prove that Binomial coefficients are log-concave? A sequence $a_0, \dots, a_n$ is log-concave if $a_k^2 \geq a_{k-1}a_{k+1}$. $$ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $$ If $ n &...
0
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3answers
44 views

If $N = ab$ and $b \geq a$ then $N\geq a^2$.

I was reading book "Higher Algebra by Barnard and Child" and got stuck on this theorem. A number N is a prime, if it is not divisible by any prime number and greater than $1$ and less than, or ...
-5
votes
3answers
40 views

Inequality troubles $\frac{x^2(5+x)(x-4)}{(x+2)(x-2)}\geq 0$ [on hold]

What $x$ satisfy the inequality: $$\frac{x^2(5+x)(x-4)}{(x+2)(x-2)}\geq 0$$ I have never been good at math, more of a right brained person. Can anybody help me to solve this inequality?I have been ...
1
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2answers
34 views

Prove or disprove this inequality

Let $p, q, a$, and $b$ be natural numbers such that $p<q$, $1<b<a$ and $b\nmid a$. Is is true that $(bp+aq)^3> (a^3+b^3)q^3$? This is what I tried: expanding the left-hand side, we ...
4
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1answer
51 views

Baby Rudin - Theorem 1.35 Cauchy Schwartz

I'm stumped on the following difficulty while reading baby Rudin (p.15). Let $A=\sum|a_j|^2, B=\sum|b_j|^2, C=\sum a_j \overline{b}_j$: $$\begin{align} \sum|Ba_j-Cb_j|^2 &= \sum(Ba_j-Cb_j)(B\...
1
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2answers
35 views

Can we say that $\prod_{i=1}^n (1-x_i)\ge 1-\sum_{i=1}^n x_i,\ \forall n\in \mathbb{N},\ \forall x_i\in [0,1)$?

The statement is easy to see to be true for $n=2,3$. However, what to do for general $n\in \mathbb{N}$? I am having this feeling that this should be a very trivial/well studied thing, but I am afraid ...
0
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1answer
24 views

Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$

I got stuck on this inequality for a day. If $p$ is positive integer, then the problem becomes too easy, but I can't find how we deal with the general case when p can be any positive number. Can ...
2
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0answers
18 views

Bernoulli inequality application

On some level of math in school we learn about Bernoulli's inequality. Proof of its correctness is very common in textbooks as exercise, when we learn mathematical induction. Is Bernoulli's ...
5
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1answer
155 views

Israel tst 2011 geometrical inequality

Inside an equilateral triangle of area $S$ lies a point, whose distances to the vertices are $x,y, z$. Prove that $xy + yz + zx \geq \frac{4}{\sqrt{3}} S$ I haven't got any idea yet. But I guess ...
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0answers
27 views

Combinatorical problem [on hold]

$k$ is a natural constant.Determine $x,y,z$ knowing that $\binom{z+k}{x+y} + \binom{z}{x} \le k$ and $2x+y \le z$.
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1answer
45 views

Elementary proofs involving inequalities

So the task of this exercise is to prove each statement. $\forall a \in$ $\mathbb R$: Prove that $a^2 \ge 0$ Does it suffice to say that $a^2 \gt 0$ or $a^2 = 0$, which means that $a \gt 0$ or $...
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0answers
15 views

Inequality with $log$ and $d^x$

Please help me to solve this inequality $Log[d] <\frac{(-1 + d^a) (d^a - d^b) b} {d^{a}(-1 + d^b) a (a - b)}$ with $0<\delta<1$, $a\geqq1$, $b>a$ and thus $b>1$. $a$ and $b$ are ...
17
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4answers
590 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 ...
36
votes
6answers
2k views

Olympiad Inequality $\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
6
votes
3answers
766 views

How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$? [duplicate]

Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$ I think this problem may be solved using nice methods, but I can't find ...
0
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0answers
12 views

Can these bounds, for the deficiency $D(x)=2x-\sigma(x)$ of a deficient number $x>1$, be improved?

Let $\sigma=\sigma_{1}$ denote the classical sum-of-divisors function. Denote the deficiency of the deficient number $x>1$ by $D(x)=2x-\sigma(x)$. Since $x>1$ is deficient, we have $D(x) \geq ...
4
votes
1answer
90 views

Show that $(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6$ for $a^2 + b^2 + c^2 + d^2 = 1$.

For $a, b, c, d \in \Bbb R$ such that $a^2 + b^2 + c^2 + d^2 = 1$, show that $$(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6.$$ The answer uses the mysterious identity $$...
14
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3answers
656 views

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $...
3
votes
2answers
59 views

Prove this inequality with trigonometry $9\cos^2{x}-10\cos{x}\sin{y}-8\cos{y}\sin{x}+17\ge 1$

let$x,y\in R$,show that $$9\cos^2{x}-10\cos{x}\sin{y}-8\cos{y}\sin{x}+17\ge 1$$ Maybe use Cauchy-Schwarz inequality can solve it?and I can't Adit it:I think the right hand can replace constant $9$ ...
2
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1answer
51 views

Number theory problem, fractions and gcd, please help!!!

The problem says "if a,b are positive integers such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer then show that $\sqrt{a+b}\ge$ gcd(a,b)" Adding $\frac{2ab}{ab}$ to $\frac{a+1}{b}+\frac{b+1}{a}$ ...
0
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2answers
39 views

How to solve $(2p_1^2-2p_1+1)^n \le 2^{-10}$ where $p_1 = 1-(1-(1/n))^N$.

Let $$S_{n,N}=(2p_1^2-2p_1+1)^n$$ and $p_1 = 1-(1-(1/n))^N$. I would like to solve $S_{n,N} \leq 2^{-10}$ for $n$. This seems hard to do exactly but is there a good approximation one can find? We ...
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1answer
63 views

Prove or disprove $x^{a_1}y^{a_2}+x^{a_2}y^{a_1}\ge x^{b_1}y^{b_2}+x^{b_2}y^{b_1}$

Prove or disprove: If $\max(a_1,a_2)\ge\max(b_1,b_2)$, then $$x^{a_1}y^{a_2}+x^{a_2}y^{a_1}\ge x^{b_1}y^{b_2}+x^{b_2}y^{b_1}$$ I can not understand it in the proof Muirhead's inequality ...
1
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2answers
51 views

Prove the inequality between integral and summation of multiplicative inverse

I want to prove the following inequality: $$ \ln(n) = \int\limits_1^n{ \frac{1}{x} dx } \geq \sum_{x = 1}^{n}{\frac{1}{x + 1}} = \sum_{x = 1}^{n}{\frac{1}{x}} - 1 $$ I ask this question as I'm ...
1
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1answer
36 views

How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B) $$ or ...
3
votes
0answers
43 views

Variant of Barrow's inequality

I proposed the conjecture as following: Let $ABC$ be a triangle, let $D$ be a point inside of $ABC$. From $D$ and $ABC$, define $F$, $E$, and $G$ as the points where the internal angle bisectors of $\...
2
votes
3answers
87 views

Prove $5a^2+b^2+c^2\geq 4ab+2ac$

I saw this question recently: Let $a,b,c$ be real numbers. Prove $5a^2+b^2+c^2\geq 4ab+2ac$. I feel like this is something with AM-GM inequality. Can someone help me with it?
5
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3answers
1k views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| \...
1
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1answer
51 views

Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
1
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3answers
36 views

Find the $\sum_{sym}ab$ maximum of the value

Let $a,b,c,d,e\in (0,1)$ and such $$a+b+c+d+e=1$$ find the maximun of the value $$S=ab+ac+ad+ae+bc+bd+be+cd+ce+de$$ I Conjecture the maximun is $\dfrac{2}{5}?$,such $a=b=c=d=e=\dfrac{1}{5}$,so $$S\...
2
votes
2answers
63 views

Prove that for any $n \ge 2$,$1\times3\times5\times \dots \times(2n-1)<n^n$ without induction

Prove that for any $n \ge 2$,$1\times3\times5\times \dots \times(2n-1)<n^n$ without induction I asked for a non induction prove but I am stuck in induction prove too. In induction we should prove ...
5
votes
2answers
66 views

An inequality in positive real continuous function

I proposed my conjecture as follows: Let $f(x)$ is a positive real continuous function that is convex on $[m, M]$, let $m \le x_i \le M$, for $i=1,2,...,n$ then show that $$\frac{f(x_1)+f(x_2)+.....+...
2
votes
3answers
44 views

Show that this inequality doesn't hold

Given $(a,b,c) \in \mathbb R^3_+$ show that atleast one of the real numbers $a(1-b)$, $ b(1-c)$ and $c(1-a)$ is less than or equal to 1\4. I tried to show it by contradiction i.e Suppose that $$a(1-...
0
votes
1answer
32 views

I'm having trouble understanding what this problem is asking me

This is the problem So my problem is that I dont know how to solve it... I have learned about system of inequalities and that kinda stuff, but I never got anything like this. I do not want anyone to ...
1
vote
10answers
117 views

A clean proof of $x^2 \geq x$, for any integer $x$

I am trying to prove that $x^2 \geq x$ for any integer $x$. Since we know that for any number $n$, $n^2 \geq 0$ we conclude that if $x \leq 0$ the proposition will hold. Next we must prove that the ...
0
votes
2answers
63 views

Solve the following using AM-GM inequality

The least value of $a \in R$ for which $4ax^2 + \frac{1}{x} \ge 1 $for all $x \gt 0 $, is Using AM-GM inequality $$\frac{4ax^2 + \frac{1}{2x} + \frac{1}{2x}}{3} \ge \sqrt[3]{a}$$ $$4ax^2 + \frac{1}{...
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votes
0answers
33 views

Hölder inequality application to show that f=1

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
5
votes
7answers
99 views

Solution of Inequality $\displaystyle \frac{1}{x-6}\le 3$

Solve the inequality: $\displaystyle \frac{1}{x-6}\le 3$ solution: \begin{align*}\frac{1}{x-6}& \le 3 \\ x-6& \le \frac{1}{3} \\x& \le 6+\frac{1}{3}\\ x&\le19/3\end{align*} but, ...
5
votes
1answer
141 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}...
1
vote
1answer
28 views

Implication of exponential growth: how is it deduced?

Let $L$ be a differentiable function defined on $\mathbb{R}\times\Omega$ with $\Omega\subseteq\mathbb{R}^n$. I will say it has exponential growth if for all $O\subset\subset\Omega$ open there exists a ...
3
votes
1answer
102 views

Minimize $P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when $xyz=1$

Find the minimum value of the product $P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when $xyz=1$ and $x,y,z$ are positive real numbers. I don't know how to go about this. AM-GM got really messy, and I don't know ...
0
votes
2answers
38 views

How to prove that ${l \choose a_1,…,a_n}\le n^{l-1} $ , when $a_1+…+a_n=l$.

In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given $a_1,...,a_n \in \mathbb{N}$ such that $a_1+...+a_n=l$, then $${l \...
2
votes
3answers
35 views

Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
0
votes
2answers
21 views

Solving the inequality involving modulus

Can I change $\frac{1}{|x-2|} \le \frac{1}{|2x-3|}$ to $|x-2| ≤ |2x-3| ? $ If I remembered correctly, I cant change $a \lt \frac{ 1}{|b|}$ to $a|b| \lt 1$ instead, I have to change it to $a-\frac{1}...