Questions on proving and manipulating inequalities.

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2
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1answer
193 views

$\inf$ and $\sup$ of a set.

Let $n\geq3$ be an arbitrarily fixed integer. Take all the possible finite sequences $(a_{1},...,a_{n})$ of positive numbers. Find the supremum and the infimum of the set of numbers ...
1
vote
2answers
61 views

How can triangle inequlity be extended to become tetrahedron inequlity?

Since each side of a tetrahedron is a triangle is it possible to have an inequality that involves triangles that can form a tetrahedron?
14
votes
4answers
655 views

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
2
votes
3answers
116 views

$\sum a_i \ln(b_i) \leq \sum a_i \ln(a_i)$ with $\sum a_i = \sum b_i = 1$

Okay, this is my another try on the question $\sup \sum a_i \ln(b_i)$ with $\sum a_i = \sum b_i =1$ which I unfortunately mis-stated and actually asked for a problem different than the one I have to ...
2
votes
1answer
58 views

$\sup \sum a_i \ln(b_i)$ with $\sum a_i = \sum b_i =1$

I was faced with the following problem: I need to know where the $\sup \sum_{i=1}^{n} a_i \ln(b_i)$ is taken if $0 < a_i, b_i < 1$ subject to the constraint $\sum a_i = \sum b_i =1$. I'm not so ...
1
vote
1answer
131 views

Find Minimum value of this expression: $P=2(a^3+b^3+c^3)+4(ab+bc+ca)+abc$

Let $a,b,c>0$ and satisfying $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\le\frac{a+b+c}{2}$. Find Min of this expression? $P=2(a^3+b^3+c^3)+4(ab+bc+ca)+abc$ From the condition, I have ...
1
vote
1answer
38 views

L^p space inequality

It's possible that this has already been answered, since I've seen this on another site. Let $a,b,c,d$ be positive functions. Is it true for all $p>1,$ that $\frac{||a+b||_p}{||c+d||_p} \leq ...
0
votes
4answers
73 views

Proving $\displaystyle\sum_{k=1}^{m+1} \frac{1}{\sqrt{k}}\gt\sqrt{m+1}$

well the original problem was to prove the sum of k to the negative one half was more that the square root of n but it thought it would be best to use induction and get the equation displayed above. I ...
3
votes
2answers
250 views

How does the triangle inequality work for $|x-y|$?

I know that $|x+y|\leq |x|+|y|$... But is it similar for $|x-y|$? That is, is $|x-y|\leq |x|+|y|$? I ask because of the following: $x-y=x+(-y)$, so $|x+(-y)|\leq |x|+|-y|=|x|+|y|$ Is it possible ...
0
votes
1answer
141 views

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$ for $|x-y|\le\alpha$.

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $$|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$$ for $|x-y|\le\alpha$. From the mean value theorem, given any $x,y$ with ...
1
vote
3answers
55 views

Simple inequality true?

For positive numbers $a,b,c,d$, is it true that $\frac{a+b}{c+d} \leq \max\{\frac{a}{c},\frac{b}{d}\}$? This question seems quite silly but I'm having a complete brain freeze..
1
vote
2answers
67 views

Prove $x^y>y^x$ by using convexity

For $y>x>e$, show that $x^y>y^x$. It is not hard to prove this inequality by using the monotonicity of $\frac{\ln t}{t}$. I am curious if this inequality can be proved by using convexity of ...
3
votes
1answer
39 views

Simple inequality question with division.

In this inequality, why do the typical rules for inequalities not hold up? $$(x/y) > 1 \quad x > y$$ However, it leaves out : $$-x < -y$$ When there is division involved in inequalities, do ...
1
vote
2answers
66 views

Inequality: if $\cos^2a+\cos^2b+\cos^2c=1$, prove $\tan a+\tan b+\tan c\geq 2(\cot a+\cot b+\cot c)$

Let $a,b,c$ be in $\left(0;\dfrac{\pi}2\right)$ such that $\cos^2a+\cos^2b+\cos^2c=1$. I am trying to prove the following inequality: $\tan a+\tan b+\tan c\geq 2\left(\cot a+\cot b+\cot c\right)$, but ...
2
votes
4answers
187 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
5
votes
2answers
109 views

Show that the sum of the series

Show that the sum of the series is greater than 24 $$\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{9}+\sqrt{11}} +\cdots+\frac{1}{\sqrt{9997}+\sqrt{9999}} > 24$$ I see ...
0
votes
1answer
109 views

If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.

I can't see any ways I could use the fact that $a+b+c=1$. I've tried solving the inequality using various AM-GM inequalities, but I just can't make it. I need some help. Thanks.
1
vote
1answer
57 views

If $a+b+c \ge 1$ and $a,b,c>0$, prove that $\frac{1}{2a+s}+\frac{1}{2b+s}+\frac{1}{2c+s} \ge \frac{1}{s}$, where $s=ab+bc+ca$.

What I know is that $$s \le \frac{(a+b+c)^2}{3} \ge \frac{1}{3}$$But as you can see the sign is pointing to different sides. So I can't see how this could be helpful. Just a small observation. I don't ...
1
vote
2answers
63 views

How prove this $(a-c)^2-4a(a+b-c)>0$

Question: let $a,b,c$ are real numbers,and such $$(2a+b)(a+b-c)<0$$ prove or disprove $$(a-c)^2>4a(a+b-c)$$ My try: since ...
0
votes
1answer
36 views

Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$. Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n ...
0
votes
2answers
26 views

Is the answer true? $\sup \frac{(\theta-\theta')^2}{\exp(n(\theta'-\theta))-1}=\frac{1}{n^2}$

I am trying to calculate the supremum ($sup$) for some equation, Is the answer true? $$\sup \frac{(\theta-\theta')^2}{\exp(n(\theta'-\theta))-1}=\frac{1}{n^2}$$
1
vote
1answer
91 views

question about an inequality in calculus [duplicate]

Please, carefully show that $$ e^{\pi} > \pi^e $$ You are not allowed to use a calculator! thanks
0
votes
2answers
120 views

Prove this inequality: $\sum{\frac{1}{a^2\sqrt{a^2+2ab}}}\ge\frac{\sqrt{3}}{abc}$

Let $a,b,c>0$. Prove this inquality: $\frac{1}{a^2\sqrt{a^2+2ab}}+\frac{1}{b^2\sqrt{b^2+2bc}}+\frac{1}{c^2\sqrt{c^2+2ac}}\ge\frac{\sqrt{3}}{abc}$
0
votes
1answer
38 views

Proving an inequality involving multiple constraints

Let $R$ be a discrete set and let $f:{\left[ {0,1} \right]^{\left| R \right|}} \times {\left[ {0,1} \right]^{\left| R \right|}} \to \mathbb{R}$ be defined as $f\left( {{\mathbf{x}},{\mathbf{y}}} ...
4
votes
1answer
101 views

Prove that :$\frac{1}{a+b} +\frac{1}{b+c} +\frac{1}{c+a}\ge \frac{4}{a^2+7} +\frac{4}{b^2+7} +\frac{4}{c^2+7}$

Let $a,b,c>0$ and satisfying $a^2+b^2+c^2=3$. Prove that :$\dfrac{1}{a+b} +\dfrac{1}{b+c} +\dfrac{1}{c+a}\ge \dfrac{4}{a^2+7} +\dfrac{4}{b^2+7} +\dfrac{4}{c^2+7}$
0
votes
1answer
26 views

Is it true that this mean inequality $ \bar{x}(m+1) \ge \bar{x}(m) $ is always correct if $m \in\mathbb Z$ and $x>0$?

Where $ \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^\tfrac1m$. I'm asking this because I've noticed that $$\bar{x}(2) \ge \bar{x}(1) \ge \bar{x}(0) \ge \bar{x}(-1).$$ Is it ...
3
votes
3answers
457 views

If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$.

The whole problem is in the title. If you wanna hear what I've tried, well, I've tried multiplying both sides by 3 and then using the homogenic mean. $${3 \over a+b+1} \le \sqrt[3]{{1\over ab}} = ...
1
vote
2answers
279 views

Mean Value Theorem to find inequality

How do I use the mean value theorem to find $$ 1+x < e^x < 1+xe^x $$ for all x>0 I'm not really sure what function I can use or if I can use any function to show with MVT. I tried using ...
3
votes
2answers
59 views

upper bound for the product of $\sin (2^k x)$.

Let $n\geq 1$ be an integer and $x$ is a real number. Prove or disprove that :$$ \left|\prod_{k=0}^n \sin\left(2^k x\right)\right|\leq\left(\frac{\sqrt{3}}{2}\right)^n.$$
4
votes
2answers
67 views

How prove this inequality $(1+\frac{1}{n})^n(1+\frac{1}{2n})>e$

let $n\in N^{+}$ show that $$\dfrac{e}{(1+\dfrac{1}{n})^n}<1+\dfrac{1}{2n}$$ My try: $$\Longrightarrow e<(1+\dfrac{1}{n})^n(1+\dfrac{1}{2n})$$ so let ...
0
votes
4answers
75 views

How to prove this ineqality

prove that $1 \leq \frac{1}{1001} + \frac{1}{1002} + ......+\frac{1}{3001} \leq \frac{4}{3} $ it seems from some Olympiad. i tried using sum of series etc. but could not get it.
0
votes
2answers
46 views

Inequality of powers

If $q$ is in $[1,2)$ and $a,b$ real numbers, is it always true that $|a+b|^q + |a-b|^q \leq 2(|a|^q+|b|^q)$ ? Thanks.
0
votes
2answers
42 views

$xz+yt\gt\frac{1}{2}(x+y)(z+t)$

Suppose $y\gt x$ and $t\gt z$ , then $$xz+yt\gt\frac{1}{2}(x+y)(z+t)$$ How can I prove this?
4
votes
4answers
119 views

Inequality $\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$

Let $f$ be a positive continuous function defined on a closed interval $[0,1]$, then it is true that: $$\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$$ I tried to show ...
0
votes
4answers
89 views

If $x>10^2$ then is the following statement true? $1-\frac{2}{x}+\frac{3}{x^2}>0.9$

If $x>10^2$ then is the following statement true? $$1\color{red}-\frac{2}{x}+\frac{3}{x^2}>0.9$$ I already figure out that $x>10^2$ implied: $$-2/x>-2\cdot10^{-2}$$ ...
2
votes
2answers
55 views

Proof of inequality with ratio of odds to evens.

A professor gave this to the class as a challenge. It may or may not have to do with calculus, but I'm running out of ideas. Prove that for any $n > 1$ $$ \frac{1}{2\sqrt{n}} < \frac{1}{2} ...
1
vote
1answer
72 views

What is the smallest possible weighted average of 1…n?

Let $a_1, ..., a_n$ be numbers in the range $[{1 \over n},1]$. Define: $$ U = \sum_{i=1}^{n}{a_i} $$ $$ W = \sum_{i=1}^{n}{i\cdot a_i} $$ I am looking for the largest possible value of the ratio $U ...
10
votes
3answers
628 views

Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$?

Which is greater among $300 !$ and $\sqrt {300^{300}}$ ? The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.
1
vote
1answer
112 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
0
votes
1answer
46 views

An inequality about conditional expectation

Let X be an non-negative random variable. Can any one tell me how to prove the following inequality? $\mathbb{P}(X>\lambda_0)\cdot \mathbb{E}[X|X>\lambda_0]\leq ...
2
votes
0answers
86 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
-1
votes
1answer
106 views

Eigenvalues of sum of Hermitian matrices

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3
votes
4answers
150 views

hard inequality i need help with

I need to show that for $x \in (0, \pi/2)$ $$ e^{\cos x} \leq (e -1) \cos x + 1 $$ I was thinking on using extrema of some function, but i have no idea. please, help me. thanks
3
votes
2answers
94 views

Is $ x \log x = O(x^{1+\epsilon})$ for every $\epsilon > 0$?

I am an amateur. Claim $$ x \log x = O(x^{1+\epsilon}) \qquad (A) $$ for every $\epsilon > 0$, $x \in \mathbb{R} \;, x > 2$. Tried to disproof this, but doubt the proof is correct. Basic ...
5
votes
1answer
68 views

function inequality $f(x+y)+y \leq f(f(f(x)))$

$f(x+y)+y \leq f(f(f(x)))$ find all possible solution for $ f: \mathbb {R} \rightarrow \mathbb {R}$
0
votes
2answers
53 views

Best constant integer inequality

Suppose $a_1,\dots,a_n$ are positive integers. Trivially one has that $$ \sum_{i=1}^n a_i^2 \leq \left (\sum_{i=1}^n a_i \right)^2 $$ I am wondering whether it is possible to make it somehow sharper, ...
5
votes
6answers
248 views

Prove that for all $x>0$, $1+2\ln x\leq x^2$

Prove that for all $x>0$, $$1+2\ln x\leq x^2$$ How can one prove that?
2
votes
2answers
299 views

Solving $x\; \leq \; \sqrt{20\; -\; x}$

This is how I tried to solve it: By squaring both sides: $x^{2}\; \leq \; 20\; -\; x$ $x^{2}\; +\; x\; -\; 20\; \leq \; 0$ Thus $-5\; \leq \; x\; \leq \; 4$ However, it seems that values less ...
0
votes
3answers
73 views

combinatorics - how many integer solutions

simple question. we are given this equation $x_1+x_2+x_3+x_4=17$ when: $0\leq x_2\leq 7$, $0\leq x_3 \leq 13$ $0\leq x_4 \leq 13$ and for all $i$: $x_i \in \mathbb Z$ we are asked how many ...
7
votes
1answer
160 views

Understanding why $\int_0^{\pi/2} \sqrt{1+\cos^2x} \geq \frac{\pi}{4}\bigl( 1 + \sqrt{2}\bigr)$

Lately I stumbled accros the magnifient paper by Roger Nelsen, which can be found here Symmetry and Integration In this paper it is shown that $$ \int_0^{\pi/2} \frac{\mathrm{d}x}{1 + ...