Questions on proving, manipulating and applying inequalities.

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1
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4answers
48 views

How to prove that $(x+c)\log(\frac{c+x}{x})>c$

How to prove that $(x+c)\log(\frac{c+x}{x})>c$ for $x, c > 0$? For $\frac{c+x}{x} \ge e$ it's obvious.
1
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0answers
19 views

Q: On Minkowski's inequality

It is well known that by applying the function $f(x)=x^p$, $0<p<1$, the inequality $$2^{1-p}(a+b)^p \ge a^p+b^p$$ holds My question is I want an equivalent form for the reverse inequality but ...
0
votes
1answer
31 views

Bounding inequalities in three dimensions

I want to write $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0$ in the form $$a \le z \le b, \quad c(z) \le y \le d(z), \quad f(y,z) \le x \le g(y,z)$$ or $$a \le z \le b, \quad c(z) \le ...
0
votes
4answers
39 views

When is it OK to assume something like $a \ge b \ge c$ when proving inequalities?

Is it okay when the inequality is cyclic instead of symmetric? For example, to prove the inequality $a^3 + b^3 + c^3 \ge a^2b + b^2c + c^2a$ (for positive real numbers $a, b, c$), can I say that WLOG, ...
5
votes
4answers
732 views

How important are inequalities?

When reading the prefaces of many books devoted to the theory of inequalities, I found one thing repeatedly stated: Inequalities are used in all branches of mathematics. But seriously, how important ...
1
vote
2answers
44 views

How do I show that $\sum_{cyc} \frac {a^6}{b^2 + c^2} \ge \frac {abc(a + b + c)}2?$

Let $a, b, c$ be positive real numbers, show that $$\frac {a^6}{b^2 + c^2} + \frac {b^6}{c^2 + a^2} + \frac {c^6}{a^2 + b^2} \ge \frac {abc(a + b + c)}2.$$ I think this is likely to turn out to be ...
3
votes
1answer
84 views

How do I show that $\frac {a^2}b + \frac {b^2}c + \frac {c^2}a \ge \frac {(a + b + c)(a^2 + b^2 + c^2)}{ab + bc + ca}?$

For positive real numbers $a, b, c$, show that $$\frac {a^2}b + \frac {b^2}c + \frac {c^2}a \ge \frac {(a + b + c)(a^2 + b^2 + c^2)}{ab + bc + ca}.$$ I don't know how to solve this at all. Can you ...
-3
votes
2answers
55 views

$s \lt t$ for each $s \in S$ and each $t \in T$. Prove that $\sup S \le \inf T$ [closed]

Let $S$ and $T$ be subsets of $\mathbb R$ such that $s \lt t$ for each $s \in S$ and each $t \in T$. Prove carefully that $$\sup S \le \inf T$$ Best way to prove such a question?
1
vote
1answer
16 views

Inequality for $x$

Consider: $\frac{x}{3} - 1 = -\frac{13}{12}$ The answer is $-3 < x < 0$. All the options are inequalities. I feel like the problem is missing something because I don't see where an inequality ...
6
votes
1answer
233 views

Understanding an example of “for all” and “for some” usage in statements.

I'm reading "Analysis I" by Tao and reviewing an appendix chapter on logic. In there he gives an example on how "for all x" is usually much stronger than just saying "for some x": "$6<2x<4$ ...
0
votes
1answer
27 views

Proving an inequality involving conditional probability

Let $(X_t)_{t\ge0}$ be a stochastic process on a probability space $(\Omega,\mathcal F, \mathbb P)$ and let $\mathcal F_t=\sigma(X_s:0\le s\le t)$. Let $\Lambda\in \mathcal F_t$ with $\Lambda\subset ...
0
votes
2answers
25 views

Show that the following inequality holds.

Suppose that n and p are positive integers. Show that $$\sum_{k=1}^{n-1} k^p < \frac{(n)^{p+1}}{p+1} < \sum_{k=1}^n k^p.$$
0
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1answer
28 views

Inequality of the expectation vs monotone function

I'm reading understanding machine learning and several of the latest lemmas I've studied involved this inequality which I've searched for but found no justification of whatsoever. Could anyone point ...
2
votes
3answers
65 views

Prove that $\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b} \leq \frac{a+b+c}{2}$

If $a,b,c \in R$, then prove that: $$\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b} \leq \frac{a+b+c}{2}$$ I can't see any known inequality working here like $H.M.-A.M.$. Could this be solved using ...
0
votes
0answers
7 views

The best possible constants such that $c_1\|a\|_{L^p}\leq \|a\|_{L^1}\leq c_2\|a\|_{L^p}$

Let $n\geq 2$ be an integer and $p$ a positive real number. Find the best-possible constants $c_1,c_2$ such that $$c_1\left(\sum_{i=1}^n a_i^p\right)^{\frac{1}{p}}\leq \sum_{i=1}^n a_i\leq ...
1
vote
1answer
35 views

How to prove $\sum_{i=1}^n |a_i|^r\leq (\sum_{i=1}^n|a_i|)^r$

I want to establish $\sum_{i=1}^n |a_i|^r\leq (\sum_{i=1}^n|a_i|)^r$, where $a_i,r \in R$ and $|.|$ is the absolute value. Is the condition $r>0$ correct? How to prove this inequality?
1
vote
1answer
78 views

Find a maximum of: $x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \cdot x $

$x,y,z \ge 0 $ , $ x+y+z =1$ Find a maximum of: $$x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \cdot x $$ and when it is reached. my attempt: 1) $$x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} ...
0
votes
2answers
29 views

Absolute value with Quadratic Inequality

This is the problem: $|x^2+2x-9|≤6$ I have no idea how to even begin with this, I'm really interested in how I should go about solving this inequality. Any help would be appreciated.
2
votes
3answers
60 views

$\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \cdots + \frac{1}{{{a_n}}} < 2$

If ${{a}_{1}},{{a}_{2}},\ldots ,{{a}_{n}}$ are distinct odd natural numbers not divisible by any prime greater than 5, then show that $\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \cdots + ...
1
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0answers
53 views

How do I show that $\frac 4{abcd} \ge \frac ab + \frac bc + \frac cd + \frac da$ for $a + b + c + d = 4$ using only AM-GM?

Let $a, b, c, d$ be non-negative numbers such that $a + b + c + d = 4$. Show that $$\frac 4{abcd} \ge \frac ab + \frac bc + \frac cd + \frac da.$$ Edit: The question already has an answer here (and ...
0
votes
0answers
34 views

If $A_1 \succeq A_2$ ,then is $A_1^{\dagger} \preceq A_2^{\dagger}$?

If $A_1 \succeq A_2$ ,then does $A_1^{\dagger} \preceq A_2^{\dagger}$ for the Moore Penrose pseudo-inverses? All matrices here are real square matrices and $\succeq$ referes to $A_1-A_2$ being ...
8
votes
3answers
175 views

How do I show that $\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge 4$ for $a^2 + b^2 + c^2 + d^2 = 4$?

Let $a, b, c, d$ be positive real numbers such that $a^2 + b^2 + c^2 + d^2 = 4$, show that $$\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge 4.$$ My try: $$\frac {a^2}b + \frac {b^2}c ...
0
votes
4answers
24 views

Finding the area under the given parameters

Q: Find the area defined by $1 < |x-2|+|y+1| < 2$ After trying a lot, I asked my friend to solve this and she got the correct answer (which is 6) by shifting the origin to (2,-1) and then ...
3
votes
1answer
62 views

Comparison of $ ( 1^a + 2^a+ … n^a)^n$ and $n^n(n!)^a $

For a given real number $a>0$ , define $ d_n =( 1^a + 2^a+ ... n^a)^n $ and $ b_n = n^n(n!)^a $ for $ n = 1,2,\ldots$ Then a) $ d_n< b_n $ for $ n> 1$, b) There exists an integer ...
1
vote
1answer
37 views

An inequality with exponents, factorials and nth roots!

Problem: Prove for natural numbers $n > 2$, $$(\sqrt{2!}-1)((3!)^{\frac{1}{3}}-\sqrt{2!})\cdots(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}) < \frac{n!}{(n+1)^n}$$. I am unable to do this one. ...
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votes
3answers
64 views

Find minimum of $a+b$ under the condition $\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$ where $m,n$ are fixed arguments

Assume $m,n \in \mathbb{R}$ is fixed. And $a,b(a>b>0)$ satisfied the equation $$\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$$ Find $\min\{a+b\}$
0
votes
0answers
24 views

using markov and chebyshev inequalities in the same problem

The problem states as following: Let's say the width in the fabrication process of an industrial metal piece has a mean of 10 and an standard deviation of 1. Find the probability for when the width is ...
0
votes
2answers
57 views

Inequality from IMO 2000 problem 4 question $\Pi_{cyc}\left(a-1+\frac{1}{b}\right)\leq 1$ $abc=1$

I know the problem is repeated but my question is somehow different. I want to know whether my proof is correct because I have troubles with the last part. Since $abc=1$ we can homogenize the ...
0
votes
1answer
34 views

proving equivalence of simple inequalities

I want to show and claim following equations hold. P is positive definite, $$ (1)\qquad x^T\cdot P\cdot x > 1 \implies x^T\cdot P\cdot x + e \gt 1, \text{ for all } e > 0\\ (2)\qquad x^T\cdot ...
0
votes
1answer
38 views

Continuity Argument?

Let $f(x)$ be a nonnegative, nondecreasing and continuous function defined on $[0,a]$. Assume that if $f(x)\le \sqrt\epsilon$ then $f(x)\le \frac{1}{2} \sqrt{\epsilon}$ holds for sufficiently small ...
1
vote
0answers
56 views

Inequality for symmetric $n \times n$-matrix with non-negative elements.

Let us consider a symmetrix $n \times n$ - matrix $A$ with non-negative elements $a_{ij} \geq 0$. Furthermore, we look at a non-negative vector $x \in \mathbb{R}^n$ with $x_i \geq 0$. Then we want to ...
-2
votes
3answers
76 views

the 53 th imo inequality

$a_2,a_3,...,a_n$ are all positive reals satisfying $a_2a_3\dots a_n=1$. Prove that $\prod_2^n(1+a_i)^i>n^n$ Please give a hint on how to approach this.
4
votes
2answers
50 views

The Set $x:\left |x+\frac{1}{x}\right|>6=?$

The question is that ,the Set $x:\left |x+\frac{1}{x}\right|>6$ equals what intervals of $x$? My approach:- I tried to solve the inequality and get interval for $x$'s value as follows:- ...
0
votes
0answers
39 views

Inequality for four numbers knowing their sum and their sum of squares [duplicate]

Let $a,b,c,d \in \Bbb R$ such that $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12$. Show that $$ abcd \le 3$$
35
votes
10answers
4k views

Why does this way of solving inequalities work?

Here is what I had to prove. Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$. Here is how my teacher did it: First assume that it is in fact, true that $a^2+b^2 \geq ...
1
vote
2answers
53 views

proving convergent sum inequality

I want to prove that for $m→∞ ⇒((1+x/m)^m→exp(x))$ My idea is to prove that there is an $m≥n$ so that $$(1+x/m)^m≥\sum_{j=0}^{n}x^j/j!≥(1+x/n)^n$$ now I would use the binomial theorem to rewrite ...
-2
votes
5answers
519 views

Help with an tricky inequality [closed]

I am struggling to prove that $$\frac{1}{N}(\exp(x)-1) > (\exp\left(\frac{x}{N}\right)-1)$$ where $x>0, N >1$. Any ideas?
0
votes
1answer
17 views

Is decimal part of inequality whole number?

If you have an inequality, x is "greater than -7" but "less than -5." Is -6 the only number that will satisfy this inequality, or will there be multiple solutions (e.g. -5.5, -6.5, etc)? In other ...
1
vote
2answers
58 views

Ratio Inequality

How can I prove that, $$\frac{a_{1}+a_{2}+\dots+a_{n}}{b_{1}+b_{2}+\dots+b_{n}} \le \max_i\left\{\frac{a_{i}}{b_{i}}\right\}$$ where $1 \le i \le n$, and $a_{i} \neq a_{j}$ and $b_{i} \neq b_{j}, ...
0
votes
0answers
30 views

Prove that either $f(-a_n)>f(0)$ or $f(a_1-a_n)>f(0)$.

I have a finite sequence of positive numbers $(a_i)_{i=1}^n$ and a function $f:[-a_n,a_1-a_n]\to\mathbb{R}$ such that ...
1
vote
3answers
36 views

Are these metrics?

I want to find if the below functions are metrics. I have worked through each of the three conditions, but am stuck on the positivity of $f(a, b)$ (first condition-see below) and the triangle ...
0
votes
0answers
22 views

How does Cauchy's inequality work?

I have the following problem: Prove $\lim_{(x,y)\to (0,0)}\frac{|x|^\alpha y^4}{x^2+y^4}$ exists. I saw elsewhere that by Cauchy's inequality that $x^2+y^4\ge2y^2|x|$. I'm not sure where this comes ...
-1
votes
2answers
115 views

Prove please that inequality ; if $x\geq 0,y\geq 0,z\geq 0$ $x+y+z\geq x^2y+y^2z+z^2x$

Prove please that inequality : if $$x\geq 0,y\geq 0,z\geq 0$$ and $$x^2+y^2+z^2=3,$$ then $$x+y+z\geq x^2y+y^2z+z^2x.$$
-2
votes
1answer
124 views

Not so easy inequality: $(x+1)(y+1)(z+1)\ge8$ [closed]

Let $x$, $y$ and $z$ be three positive numbers such that $x+y+z=xy+xz+yz$ prove that $(x+1)(y+1)(z+1)\geq8$
1
vote
0answers
23 views

Poisson process deviations

How can one prove the following inequalities for a standard Poisson process $\mathbf{N}(t)$ ? $\mathbb{P}\bigg[\bigg|\frac{\mathbf{N}(\lambda)}{\lambda}-1 \bigg| > \varepsilon\bigg] \leq ...
1
vote
2answers
40 views

If $a,b,c>0\;,$ Then value of $\displaystyle \lfloor \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\rfloor $,

If $a,b,c>0\;,$ Then value of $\displaystyle \bigg \lfloor \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\bigg\rfloor $, Where $\lfloor x \rfloor $ Rep. floor function of $x$. $\bf{My\; Try::}$ ...
0
votes
2answers
19 views

How to give a bound e.g $\leq \epsilon$ in $e^{\frac{M^2 \log_e (\epsilon) + M^2 \log_e^2 (\epsilon)}{2\delta}}$

$\epsilon$ in $$e^{(M^2 \log_e (\epsilon) + M^2 \log_e^2 (\epsilon))/(2\delta)}$$ I know that $e^{M^2\times \log_e \epsilon}$ would result in $\epsilon^{M^2}$ But I am confused what to do with ...
0
votes
1answer
51 views

How to show that $a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd$ if $a,b,c,d>0$

How can I prove that if $a,b,c,d>0$ then $$a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd?$$ I think there is some simple proof but I can't remember... is this a special case of some general inequality? ...
2
votes
1answer
45 views

A conditional inequality which itself implies a sharper version of it [duplicate]

Problem: Given that $m, n$ are positive integers such that $\sqrt{7} -\frac{m}{n} > 0$. Then show that $\sqrt{7}-\frac{m}{n} > \frac{1}{mn}$. I have failed to do this fascinating problem. My ...
2
votes
0answers
24 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...