Questions on proving and manipulating inequalities.

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17 views

How to solve integer program using MIR inequalities?

How do you solve an integer program using Mixed Integer Rounding inequailities? Since it's a rounding inequality I am confused how to approach this when the coefficients of the problem are all ...
5
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1answer
80 views

Inequality with sum of Binomial coefficients.

Prove that for every positive integer $n \ge 2$$$\sum^n_{k=1}k \sqrt{\begin{pmatrix}n\\ k\end{pmatrix}}\leq\sqrt{2^{n-1}n^3}$$ I tried it by induction but I didn't know how to end it.
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1answer
31 views

Spin the bottle: Comparing 2 Euler angles

Angle A is a Euler angle that keeps increasing by increments and Angle B is the stopping point of Angle A (think 'Spin-the-bottle' where Angle A is the current angle of the bottle Angle B is the angle ...
5
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4answers
66 views

How to proof that $\sum_{i=1}^{2^n} 1/i \ge 1+n/2$

I had troubles trying to prove that for every $n\ge1$ $$\sum_{i=1}^{2^n}\frac1i\ge 1+\frac n2$$ Can you give me a hint about the induction proof or show me in detail how can I prove it? I would ...
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3answers
72 views

Find the maximum value for $x+y+z-xy-yz-zx$

If $x,y,z$ are real numbers for which holds $0\le x,y,z \le 1$, then find the maximum value of $x+y+z-xy-yz-zx$ and find $(x,y,z)$ for which you get the maximum value. This is how did it and I would ...
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1answer
44 views

What inequality was applied?

I'm reading some probabilistic paper and have a trouble with understanding some part. Here is this part: mu is a Lipschitz function and M is the Lipschitz constant, and: What inequality was ...
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0answers
37 views

Proof using Archimedean property and Bernoulli's inequality

I am trying to prove the theorem below (using both the Archimedean property and Bernoulli's inequality). As usual, I would like to write a highly intelligible proof. Any constructive feedback is ...
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1answer
76 views

Prove inequality for every positive integer

Prove that for every positive integer $n$ : $ \frac {1}{3}+ \frac{2}{3\cdot5}+\ldots+ \frac{n}{3\cdot5\cdot \ldots \cdot(2n+1)}<\frac{1}{2}$. I see sum is $\frac ...
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1answer
68 views

If $0 \leq a, b, c \leq 1$, the inequality $| a - b + c| \cdot |(a - c)^2 + b^2| \leq 1$ holds

Please, help me to prove the inequality or find a counter-example For $0 \leq a, b, c \leq 1$, prove that $|a - b + c| \cdot |(a - c)^2 + b^2| \leq 1$.
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1answer
34 views

Inequality involving binomial coefficients

I recently stumbled upon an inequality involving binomial coefficients. There is reason to suspect that it holds for all $l\in\mathbb{N}$. It states that $$ (2l+1)^{2l+1} < \sum_{m = 0}^{l} ...
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1answer
43 views

To find positive roots of a quadratic equation

Find all values of $a$ so that: $$x^2- ax+(4a+1)= 0$$ has both roots positive. I have been working hard and long on known facts but am unable to crack this one. Finally saw this site and hope for ...
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1answer
57 views

Relation between Sum and Product of Matrix Eigenvalues

For any square matrix $A$, let $\alpha $ be the smallest eigenvalue of $% \frac{1}{2}\left( A^{T}+A\right) $, and let $\beta $ be the largest eigenvalue of $A^{T}A$. Does the following relationship ...
2
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1answer
68 views

An inequality for matrix

$A$, $B$ are Hermitian matrix, how to show $$|\operatorname{trace}(e^A)|\geq|\operatorname{trace}(e^{A+iB})|$$ As a special case, if A is zero, then the inequality is reduced to $n\geq ...
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0answers
35 views

Is there an upper bound for the following rational expression, in terms of $b^2$?

In the following, $b$ and $r$ are both positive integers. Is there an upper bound for the following rational expression, in terms of $b^2$? $$U = \frac{2{b^2}\left({b^2}{2^r}({2^r} - 1) + 2^{r+1} - ...
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2answers
101 views

Show that $\bigl| e^x + e^{-x}-2-x^2\bigr| \le {x^4 \over 6} $ for $|x| \leq 1$

My try at it $$ \left| e^x + e^{-x}-2-x^2\right| \iff | f(x) - p_2(x)| = |R_3(x)| $$ where $ f(x) = e^x + e^{-x} $ and $ |x| \le 1 $ This gets me $$ |R_3(x)| \le (e-e^{-1}) {x^3 \over 6} $$ This ...
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4answers
47 views

Prove that for all $y,z\in\mathbb{R}^+$ it is true that $(y + z) (1 - 8 y z + z^2 + y^2 (1 + 9 z^2))\ge0$

Prove that for all $y,z\in\mathbb{R}^+$ it is true that $$(y + z) (1 - 8 y z + z^2 + y^2 (1 + 9 z^2))\ge0$$ It is obvious that $y+z>0$. Then I tried to reduce $1 - 8 y z + z^2 + y^2 (1 + 9 z^2)$ ...
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1answer
36 views

$0 \leq x \leq \dfrac{\epsilon}{1+\epsilon} \implies 1\leq \left(1+\dfrac{x}{n} \right)^{n}\leq \epsilon$

show that $\forall n\geq 1,\quad \forall \epsilon > 0,\forall x \in \mathbb{R},$ $$0 \leq x \leq \dfrac{\epsilon}{1+\epsilon} \implies 1\leq \left(1+\dfrac{x}{n} \right)^{n}\leq ...
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3answers
75 views

$1\le \bigl(1+\frac{\alpha}{n}\bigr)^{n}\le \frac{1}{1-\alpha}$

Let $n\geq 1$ and $\alpha \in [0,1)$ show that : $$1\le \left(1+\dfrac{\alpha}{n}\right)^{n}\le \dfrac{1}{1-\alpha}$$ This question is related to that one ${n \choose k}\leq n^k$ My thoughts: ...
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0answers
28 views

Gigliardo-Nirenberg-Sobolev inequality for functions in $W^{k,p}$, without zero trace.

The G-N-S inequality can be stated as follows: Let $U\subset\mathbb R^d$, open bounded, with $C^1$ boundary, then for any $w\in W^{k,p}_0(U)$, $p<d$ $$\|w\|_{L^{p^*}(U)}\le C(d)\|\nabla ...
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3answers
100 views

${n \choose k}\leq n^k$

Let $n$ et $k\in \mathbb{N}$ such that : $k\leq n $ Show that :$${n \choose k}\leq n^{k}$$ My thoughts: note that for all $\ k\leq n$ : $${n \choose k}=\frac{n!}{k!(n-k)!}$$ To prove ...
5
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2answers
72 views

Minimum of $f(z) = \left|z^2+z+1\right|+\left|z^2-z+1\right|$

How Can I calculate Minimum value of $f(z) = \left|z^2+z+1\right|+\left|z^2-z+1\right|\;,$ Where $z = x+iy$ and $i=\sqrt{-1}$ $\bf{My\; Try::}$ Let $z= x+iy\;,$ Then $z^2+z+1 = ...
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1answer
51 views

AM-GM Inequality Problem involving Factorial

Problem: Prove that $\sqrt{n}\le (n!)^{1/n}$ for every positive integer $n$. I know that the AM-GM inequality is involved in this but I don't exactly see how. However, I do see that $(n!)^{1/n}\le ...
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1answer
38 views

Analytical proof of the inequality $p^n (1-p\ln p)<1$ where $0<p<1$

Simulations with Mathematica suggest to me that $$p^n (1-p\ln p) <1\quad\text{for } p \in (0,1) \text{ and }n \ge 0$$ Have you got any hints on how I can derive an analytical proof of this ...
2
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2answers
46 views

Trigonometric inequality $|\sin{a_1}|+|\sin{a_2}|+…+|\sin{a_n}|+|\cos{(a_1+a_2+…+a_n)}| \ge1$ for all real $a_i$

Prove that for all real numbers $a_1,a_2,...,a_n$ the following inequality holds: $$ |\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1 $$
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1answer
34 views

Inequality with a differentiable function + diffeomorphism

Assume that $h:\mathbb{R} \to \mathbb{R}$ is a differentiable function for which there is a number $\lambda \in \mathbb{R}_+^n$ so that: $$\lvert ((dh)(x)(t)\rvert \ge \lambda \lvert t \rvert, \forall ...
3
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0answers
44 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
4
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1answer
68 views

How prove this inequality $(y-z)^2>8(y+z)$

For three distinct postive integer $x,y,z$ such $$(x+y)(x+z)=(y+z)^2$$ show that $$(y-z)^2>8(y+z)$$ My some idea: since $$x^2+(y+z)x+yz-(y+z)^2=0$$ so ...
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6answers
130 views

Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$

Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$ I seem to struggle with this simple proof. All I managed to find is that ac+bd=-4 (which might not even be correct).
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1answer
63 views

How to show that $\lim\limits_{t \rightarrow \infty} S(t) = \Lambda/\mu$?

Can someone here help me establish that $\lim\limits_{t \rightarrow \infty} S(t) = \Lambda/\mu$, given that: $\frac{dS}{dt}=\Lambda-\mu S-\beta \frac{S}{N}(H+C+C_1+C_2)-\tau \frac{S}{N}(T+C)$ ...
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1answer
37 views

Equality case in elementary form of Holder's Inequality

A well known elementary formulation of Holder's Inequality can be stated as follows: Let $a_{ij}$ for $i = 1, 2, \dots, k; j = 1, 2, \dots, n$ be positive real numbers, and let $p_1, p_2, \dots, p_k$ ...
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1answer
151 views

If $ab+bc+ac=3abc$ then prove that $a^2b+b^2c+c^2a\geq 2(a+b+c)−3$

I have this problem and I tried to do it but I did too much calculations and I feel like I'm using the wrong method. Here's the problem : $a,b,c$ are all real and positive numbers. If ...
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1answer
37 views

$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem Let $x,y,z \in ...
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1answer
43 views

Prove Bernoulli inequality if $h>-1$

Qi) Prove Bernoulli's inequality If $h> -1$, then $ (1+h)^n \geq 1+nh$ Qii) why is this Trivial is $h>0 $ Something i have always been lucky with is having a lot of intuition to go on with ...
1
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1answer
70 views

How to prove that there exist $m,n,p,q\in [1,n]$ such that $\text{Icm}[a_{m},a_{n},a_{p},a_{q}]\ge n^2$

Question: let $a_{1},a_{2},a_{3},\cdots,a_{n}(n>100)$ be quie different postive integers,and suc for any quite different postive integer $i,j,k,l\in [1,n]$,have ...
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1answer
95 views

Is it true that $\cos(x^2) \leq \cos(x \log x) \leq \cos(x)$? [closed]

Why is this inequality true? $\cos(x^2) \leq \cos(x \log x) \leq \cos(x)$. Thanks,
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1answer
67 views

Convex function property

Let $ f_{1}, f_{2},..., f_{n} $ convex functions in the interval $[0,1]$ such that $ max(f_{1},f_{2},...,f_{n}) \geq 0 $ show that there exist positive real numbers $a_{1}, a_{2},...,a_{n} $ not ...
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0answers
26 views

$ \|u(x)+u(x)-x +o(\|x\|^p)\|<r $?

Set $B_r=\{ x\in \mathbb{R}^n : \|x-0\|< r \}$ for any $r>0$. Let $C^p_0(B_r,B_r)$ the set of all smooth functions $u:B_r \to B_r$ of class $C^p$ such that $u(0)=0$. I would like to prove the ...
3
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2answers
173 views

Use AM-GM to prove upper bound.

While studying for my upcoming exams I came across a problem in the AM-GM section: If $a_n = (1+\frac{1}{n})^{n}$ , $n \in \mathbb N$ then prove that: $$2 < a_n < 4$$ Proving the ...
4
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3answers
91 views

Inequality for $C^1$ function: $|f(x)|^2 \le \frac{1}{2}\tanh \frac{1}{2}\int_0^1 (|f(x)|^2+|f'(x)|^2)\,dx$

Prove for $f\in C^{1}[0,1]$ such that $f(0) = f(1) = 0$, the following inequality: $$|f(x)|^2 \le \left(\frac{1}{2}\tanh \frac{1}{2}\right)\left(\int_0^1 (|f(x)|^2+|f'(x)|^2)\,dx\right)$$ This is ...
2
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1answer
73 views

An AM-GM inequality for the harmonic numbers

If $H_n=1+\frac 12+\frac 13+...+\frac 1n(n>2)$, prove that $$n(n+1)^{1/n}-n<H_n<n-(n-1)n^{-\frac 1{n-1}}$$ I did the following $$\frac {1+\frac 12+\frac 13+...+\frac 1n}n\ge \frac ...
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0answers
47 views

Nonhomogenous Inequality, symmetric

Show that $3Z+\left(\dfrac{3-X}{3Y-X} \right)^3+\dfrac{X(3-X)}{3Y-X} \ge 5$, where $X=abc,$ $Y=\dfrac{ab+bc+ca}{3},$ $Z=\dfrac{a^3+b^3+c^3}{3}$, for positive real $a$, $b$, and $c$, such that $abc ...
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1answer
30 views

Simplifying Inequality Involving $\sigma$, $\beta$ and $x$

Given that $\sigma>0$, $\beta>0$, $x>0$ and $\sigma>\beta$, there are a couple of simplifications I cant derive: $$1.\,\,\sigma \geq x\,\,\,\,and\,\,\,\,\sigma\beta\geq ...
1
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3answers
110 views

Find Minimum Value of: $P=x^2+y^2+2xy+2-\frac{1}{xy}$

Given: $x,y>0$ and $x^2y-x+xy^2-y-3xy=0$ Find Minimum Value of: $P=x^2+y^2+2xy+2-\frac{1}{xy}$ I found $\min P =\frac{71}{4}$ at $x=y=2$ but I cant prove that Could some one help me ?
2
votes
3answers
56 views

Solve $x+a^x<b$ algebraically

The answer to $x+3^x<4$ is $x<1$ by plotting the graph of $y=x+3^x$ and $y =4$. Is there a way to get to the solution algebraically? Updated: is there a way to get to the solution of ...
1
vote
2answers
66 views

Is there a lower bound for the following rational expression, in terms of $b^2$?

In the following, $b$ and $r$ are both positive integers. Is there a lower bound for the following rational expression, in terms of $b^2$? $$L = \frac{b^2\left(2^{r+1} b^2(2^r-1) + 3\cdot 2^r - ...
3
votes
1answer
119 views

Proving the inequality $ \left|\prod_{i=0}^n \left(x - \frac{i}{n}\right)\right| \le \frac{n!}{4n^{n+1}}$

Let $n \in \Bbb{N}$ and $x \in [0,1]$ prove $$ \left|\prod_{i=0}^n \left(x - \frac{i}{n}\right)\right| \le \frac{n!}{4n^{n+1}}$$ I manage to show that $\left| (x-\frac{n-1}{n})(x-\frac{n}{n})\right| ...
0
votes
0answers
36 views

inequality for point on a sphere

Question: Let $a,b,c,d$ be positive real numbers which satisfy $a^2 + b^2 + c^2 + d^2 = 1$. Define $f(a,b,c,d) := \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}$. Show that $$ f(1-a,1-b,1-c,1-d) \geq ...
2
votes
3answers
76 views

Let $a,b \in \mathbb R$ and $f(x)=a\cos x+b\cos3x$. Prove that $|b|\le 1$.

Let $a,b \in \mathbb R$ and $f(x)=a\cos x+b\cos3x$. It is known that $f(x)>1$ has no real solutions. Prove that $|b|\le 1$. We can write the given equality as \begin{align} &f(x)=(a-3b)\cos ...
2
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1answer
35 views

Cubic inequality

If $x,y,z$ are reals from $[0,1]$, then prove that $$2(x^3+y^3+z^3)-x^2y-y^2z-z^2x \le 3$$ We can assume $x=sin\theta, y=sin\phi,z=sin\gamma$. Therefore inequality can be written as ...