Questions on proving and manipulating inequalities.

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2answers
109 views

How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China ...
0
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1answer
48 views

How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$?

How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$ where $x, y\geq 1$?
0
votes
0answers
61 views

equivalent LMIs

page 63 of LMI book of Stephen boyd: why and how LMI conditions 5.14 and 5.12 are equivalent? how to get from 5.12 to 5.14? [ A'P+PA+Lambda*C'*C PB+lambda*C'*D; (PB+lambda*C'D)' -lambda(I-D'*D)] ...
9
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11answers
408 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
1
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2answers
27 views

A question about an inequality involving integers $\ge 0$ [closed]

$y^2 \le x < (y + 1)^2$ is true for a unique $y$ where $y \ge 0$ and $x \ge 0$ . Suppose $x > 9$, then why is $y \ge3$? Suppose $x = 10$. Then $y = \sqrt{10}$ or $y < \sqrt{10}$. So, ...
0
votes
1answer
25 views

Chong inequalites about permutations

I read about two inequalities called Chong's inequalities. They state: $$\sum_{k=1}^N\dfrac{a_k}{a_{\pi(k)}}\ge N$$ and $$\displaystyle\prod_{k=1}^Na_k^{a_k}\ge\prod_{k=1}^N a_k^{a_{\pi(k)}}$$ I ...
0
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2answers
30 views

Product across inequalities

Suppose I have a sequence of numbers $\{a_i \}_{i=1}^{N}$ and $\{b_i \}_{i=1}^{N}$ with $0\leq a_i\leq 1$ and $0 \leq b_i \leq 1$ for $i=1,...,N$. Is stating If $a_i \leq b_i$ $\forall i$ then ...
1
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2answers
67 views

$a,b,c \geq 0$,prove that $a^2+b^2+c^2+abc+5 \geq3(a+b+c) $

$a,b,c \geq 0$, prove that $a^2+b^2+c^2+abc+5 \geq3(a+b+c)$ I'm certain that this problem could be solved by using dirchlet's theory.but I do not know how to apply it exactly.
4
votes
1answer
68 views

Prove $a+b+c \geq ab+bc+ca$

If $a,b,c$ are positive real numbers that $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\geq 1$$ is true,Prove:$$a+b+c \geq ab+bc+ca$$ Additional info:Additional info: We should only ...
1
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1answer
60 views

Proof of an inequality involving $(N-1)!$

How is it possible to prove the following inequality? ...
3
votes
2answers
86 views

How find the maximum of the value $x^2_{1}+x^2_{2}+\cdots+x^2_{2014}$

Question: let $x_{i}\in[-11,5],i=1,2,\cdots,2014$,and such $$x_{1}+x_{2}+\cdots+x_{2014}=0$$ find the maximum of the value $$x^2_{1}+x^2_{2}+\cdots+x^2_{2014}$$ since ...
5
votes
2answers
127 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq ...
1
vote
1answer
31 views

Computational complexity and the big $\mathcal{O}$

I have a question about this Big $\mathcal{O}$ problem. I have the question down $90\%$, but the other $10\%$ isn't getting to me. I will write out the entire question and I'll point out the step, ...
5
votes
4answers
361 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
5
votes
3answers
248 views

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal ...
4
votes
8answers
187 views

Show that the inequality holds $\frac{1}{n}+\frac{1}{n+1}+…+\frac{1}{2n}\ge\frac{7}{12}$

We have to show that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$ To be honest I don't have idea how to deal with it. I only suspect there will be need to consider two ...
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3answers
77 views

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=1$ prove that $a+b+bc+ac-abc \leq 2$

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$ can any one help me with this problem,I believe Dirichlet's theorem is the key for this sorry for making mistake over and over ...
1
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2answers
67 views

Induction inequality on sum of reciprocals

I have to prove that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{1}{2}$ for natural $n$ Checking for $n=1$ we have $\displaystyle 1+\frac{1}{2}=\frac{3}{2}\ge \frac{1}{2}$ ...
2
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1answer
100 views

Prove $\sum \limits_{cyc} \frac{1+b^2+c^4}{a^+b^2+c^3}\geq 3$

If $a,b,c$ are positive real numbers, prove $$\sum \limits_{cyc} \frac{1+b^2+c^4}{a+b^2+c^3}\geq 3$$ Additional info: We should only use Cauchy (preferred to used at least once and more than ...
1
vote
1answer
58 views

To control first derivative with the function itself.

Let $f$ be a compactly supported positive $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2< C f(x) $ by showing that for every point ...
1
vote
3answers
27 views

Inequality involving floor function and fractions

I have little to no experience working with floor inequalities so I am kind of stuck on this one. It seems pretty intuitive though. So basically I want to show that ...
4
votes
2answers
97 views

prove $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq 3(a^2+b^2+c^2)$

If $a,b,c$ are positive real numbers and $a+b+c=1$,Prove: $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq 3(a^2+b^2+c^2)$$ Additional info:We can use AM-GM and Cauchy inequalities mostly.We are ...
4
votes
2answers
58 views

prove $\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$

If $a,b,c$ are positive real numbers,prove:prove $$\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$$ Additional info:We can use AM-GM and Cauchy ...
0
votes
0answers
36 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
0
votes
1answer
43 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
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1answer
53 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
1
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0answers
51 views

Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...
10
votes
1answer
138 views

prove that : $\frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c}+ \frac{c^2+a^2}{c+a} \geq 3$

For $a^2+b^2+c^2 =3$, with $a,b,c$, positive real numbers, prove that $$\frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c}+ \frac{c^2+a^2}{c+a} \geq 3.$$ can any one help me with this problem.
1
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1answer
35 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
2
votes
1answer
53 views

prove $\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$

If $x$,$y$,$z$ are positive real numbers,Prove:$$\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$$ Using this two inequality: $\sum ^n_{i=1} \sqrt{a_ib_i}\leq\sqrt {ab} $ (we call it ...
6
votes
3answers
178 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
2
votes
2answers
121 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
2
votes
3answers
110 views

How to prove $ \frac{e^{x}+e^{-x}}{2} \le e^{\frac{x^2}{2}} $?

Let $x\in \mathbb{R}$, how to prove $$ \frac{e^{\large x}+e^{\large-x}}{2} \le e^{\large\frac{x^2}{2}} $$
3
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4answers
57 views

prove $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

$a,b,c$ are positive real numbers.prove:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info:we cant use induction.we should mostly use Cauchy inequality.other ...
2
votes
0answers
50 views

$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $ [closed]

Show that for every arbitrary complex number a,b and c we have $$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $$ Thanks.
1
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0answers
83 views

Practicing the arithmetic-geometric means inequality

I am struggling with learning the AM-GM Inequality that is considered vital to know for math olympiads, contests, etc. I just don't really know when to use it, when it is necessary to use, the purpose ...
2
votes
6answers
231 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
6
votes
2answers
153 views

prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$

If $a_i$ and $b_i$ are positive, and $b= \sum ^n_{i=1} b_i$,$a= \sum ^n_{i=1} a_i$ prove $$\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$$ Additional: we should just use Cauchy ...
2
votes
1answer
39 views

Prove $\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$

If $\space x_1+x_2+\cdots+x_n=1$ and all $x_1,x_2,\cdots,x_n$ are positive and real numbers, prove:$$\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$$ ...
0
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4answers
51 views

Prove that $r(n+1−r) \ge n$ for any positive integer $n$ and $1 \le r \le n$. [closed]

Prove that $r(n+1−r) \ge n$ for any positive integer $n$ and $1 \le r \le n$. Nothing I tried worked, because the coefficient of $r$ is negative so I only get maximums and not minumums.
4
votes
1answer
112 views

Deriving inequalities featuring bounded variables

I have a model which fits certain thermodynamic data, of the form $$y = \frac{ax}{ 1 + (a - 1)x} + bx(1 - x) \quad a,b \in \mathbb{R} \quad 0 \leq x \leq 1$$ Thermodynamics dictate that ...
1
vote
1answer
51 views

Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
3
votes
3answers
127 views

Prove that $n^n \le (n!)^2$.

Prove that $n^n \le (n!)^2$. There is an elementary solution, which I haven't been able to find. So far I tried manipulating and pairing terms but nothing worked. I would appreciate any help!
4
votes
2answers
110 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
2
votes
1answer
175 views

Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
4
votes
2answers
69 views

How to prove that $ 1- \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1-\frac{x}{n})^n$

How would I prove this inequality (assuming its true, its from a textbook) $$1 - \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1+\frac{-x}{n})^n$$ if $n > |x|$, $x\in R$ and $n\in N$ I first ...
3
votes
0answers
48 views

How prove $\max\lbrace \cot{A}(\cot{Y}+\cot{Z}),\cot{B}(\cot{Z}+\cot{X}),\cot{C}(\cot{X}+\cot{Y})\rbrace\ge\frac{2}{3}$

let $\Delta ABC,\Delta XYZ$ are acute triangle show that $$\max\lbrace\cot{A}(\cot{Y}+\cot{Z}),\cot{B}(\cot{Z}+\cot{X}),\cot{C}(\cot{X}+\cot{Y})\rbrace\ge\dfrac{2}{3}$$ My idea: since ...
0
votes
0answers
22 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
1
vote
1answer
61 views

Does the following sequence converge?

Suppose $a_i>0$ for all $i$, $\frac{\sum_{i=1}^n a_i}{n}\to \infty$ and p>1. Let $$y_n = \frac{(\sum_{i=1}^n a_i)^p}{n^{p-1}\sum_{i=1}^n(a_i^p)}.$$ Is $y_n$ monotonic? How can you prove or disprove ...
6
votes
3answers
168 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...