Questions on proving, manipulating and applying inequalities.

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0
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2answers
36 views

Logarithmic inequality: $\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$

I need help solving this: $$\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$$ So far I could not make sense of this, because I don't understand how to handle $\log^2$ or the $+6$ at ...
2
votes
1answer
50 views

$sup_{x,y\in R}{(\cos{x^2}+\cos{y^2}-\cos{xy})}-\inf{(\cos{x^2}+\cos{y^2}-\cos{(xy)})}=6$

let $x,y\in R$,prove or disprove $$sup_{x,y\in R}{(\cos{x^2}+\cos{y^2}-\cos{xy})}-\inf_{x,y\in R}{(\cos{x^2}+\cos{y^2}-\cos{(xy)})}=6$$ I think we must show that ...
2
votes
1answer
188 views

Prove $\liminf a_n + \liminf b_n\le \liminf (a_n +b_n) $ [duplicate]

$a_n$ and $b_n$ are two bounden sequences Prove $$\liminf(a_n) + \liminf(b_n)\le \liminf(a_n +b_n)$$ Should I use $$\inf(a+b) = \inf(a) + \inf(b)$$ and i could not come up with how to proceed from ...
1
vote
3answers
49 views

Show that $2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$

If $a,b,c$ are positive real numbers, not all equal, then prove that $$2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$$ How can I show this?
2
votes
1answer
143 views

Is $\frac{1}{e^\gamma\log x} \prod\limits_{p < x,p\,\text{prime}} \frac{p}{p-1}<1+ \prod\limits_{p<x,p\,\text{prime}}\frac{1}{p^{n+1}-1}?$

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with ...
1
vote
1answer
50 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
3
votes
2answers
51 views

Maximising a sum - closed form?

As a follow up to this question, I am wondering the following: Suppose $\sum_{i=1}^n x_i=0,\;\sum_{i=1}^n x_i^2=1$. Is it there a closed form for $\max \sum_{i=1}^n x_ix_{i+1}?$ ($x_{n+1}=x_1$) For ...
2
votes
7answers
162 views

Irrational number inequality : $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}>\sqrt{3}$

it is easy and simple I know but still do not know how to show it (obviously without simply calculating the sum but manipulation on numbers is key here. ...
9
votes
1answer
125 views

Inequality for $N(0,1)$ CDF: $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does ...
1
vote
1answer
64 views

Proving Cauchy-Schwarz related proof using induction

So the first thing I was asked to prove was this: If $a_1,a_2,...,a_n$ and $b_a,b_2,...,b_n$ are real numbers, use induction to show. ...
1
vote
0answers
34 views

Inequality about squareroots [duplicate]

If $a,b\geq 0$ show that $\left| \sqrt{a}-\sqrt{b}\right|\leq\sqrt{\left|a-b\right|}$. WLOG we can assume that $a\geq b$. If one of them is $0$ this is trivial. So assume none of them is $0$. Now, ...
0
votes
1answer
25 views

Inequality involving different diameter average

I have found an assertion in a scientific book (Hinds, Aerosol Technology, 2nd Edition, 1998, p. 83-84) that claims: Given the general form [here for grouped data] for the diameter of an average ...
1
vote
1answer
56 views

Maximisation problem

I am trying the following question: If$$a+b+c+d=0,\;a^2+b^2+c^2+d^2=1$$ Then what is the maximum value of $ab+bc+cd+da?$ By the rearrangement inequality I can get $ab+bc+cd+da\leq 1$ but I am ...
2
votes
0answers
63 views

On the average length of the Steiner net for $n$ randomly chosen points in the unit square

$n$ points are randomly chosen in the unit square with respect to the uniform measure. What is the average length $L$ of the associated Steiner net (tree of minimum length through each of the $n$ ...
0
votes
2answers
111 views

What's the name of this strange inequality?

There is an inequality: $$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$ which I even don't know its name. I'd like to have an ask ...
5
votes
1answer
131 views

Prove, using the method of mathematical induction that the following holds true

For natural numbers $n\ge1$ show the following inequality using induction. $$n^{1/n}\le 1+\sqrt{\frac{2}{n}}$$
0
votes
2answers
46 views

Inequality about sum of finitely many real numbers

Suppose that $a_1, a_2, ..., a_n$ are non-negative real numbers. Put $S = a_1 + a_2 + ... + a_n$. If $S < 1$, show that $$ 1+S\leq(1+a_1)...(1+a_n)\leq\dfrac{1}{1-S}.$$ I tried induction on $n$ ...
0
votes
1answer
25 views

Unclear Application of Cauchy's Inequality

I was looking for a solution to a problem (both found here), where I came across the following ($a, b, c > 0$): Applying Cauchy's inequality, we get $(\frac{c}{a+2b} + \frac{a}{b+2c} + ...
1
vote
1answer
161 views

upper bound on this matrix norm

What would be the upper bound on the 2-norm (or any norm) of the following matrix product ? Please consider the smallest upper bound. $\|\left(I+BA^T\right)\left(I+AA^T\right)^{-1}\|< ?$ where A ...
2
votes
1answer
40 views

An elementary inequality about $n$-th roots

I want to show that for each $m,n\in\Bbb{N}$, $$\large{ \dfrac{1}{\sqrt[n]{1+m}}+\dfrac{1}{\sqrt[m]{1+n}}\geq 1}.$$ I tried induction but it doesn't work. Tried to apply the Bernoulli inequality ...
1
vote
1answer
28 views

$a^{x}\left( y-z\right) +a^{y}\left( z-x\right) +a^{z}\left( x-y\right) >0$

If a is any positive number except $1$ , and $ x, y, z,$ are REALS no two of which are equal, then $a^{x}\left( y-z\right) +a^{y}\left( z-x\right) +a^{z}\left( x-y\right) >0$. It is quite easy ...
2
votes
3answers
98 views

Prove $x \geq \sin x$ on $[0,\pi/4]$

As the title says.. it says to use the mean value theorem but I don't see how that's applicable. Thank you
2
votes
4answers
79 views

Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $

I have the following inequality on my class notes that I haven't been able to prove, I was even wondering if it is actually true: $$ \forall a,b \in \mathbb{R}^{\ge0} \left( \left| \sqrt{a}-\sqrt{b} ...
2
votes
2answers
46 views

Inequality using only algebraic ''moves''

How can I verify the following inequality using only algebraic passages? $$ 5^\frac{1}{3} + 6^\frac{1}{2} > or < 4 $$
2
votes
2answers
37 views

Help in proving inequality of arbitrarily arranged numbers.

Let $p_1, p_2, p_3,.....p_n$ be an arbitary arrangement of natural numbers from $1$ to $2014$. Prove that $$\frac{1}{p_1+p_2} + \frac{1}{p_2+p_3} + \frac{1}{p_3+p_4} + ... + ...
1
vote
1answer
92 views

Inequalities - proof by induction that $n^2 \leq n!$ for $n\geq 4$

Proof by induction involving inequalities completely escapes me. I've encountered the following problem: For which non-negative integers n is $n^2 ≤ n!$? Prove your answer (by induction). So, ...
1
vote
5answers
136 views

$x,y,z$ are positive real numbers and $x+y+z=1$ $\implies$ $\bigg(1+\dfrac 1x\bigg)\bigg(1+\dfrac 1y \bigg)\bigg(1+\dfrac 1z \bigg)\ge 64$?

If $x,y,z$ are positive real numbers such that $x+y+z=1$ then is it true that $\bigg(1+\dfrac 1x\bigg)\bigg(1+\dfrac 1y \bigg)\bigg(1+\dfrac 1z \bigg)\ge 64$ ?
4
votes
5answers
150 views

Prove $(b-a)\cdot f(\frac{a+b}{2})\le \int_{a}^{b}f(x)dx$

Let $f$ be continuously differentiable on $[a,b]$. If $f$ is concave up, prove that $$(b-a)\cdot f\left(\frac{a+b}{2}\right)\le \int_{a}^{b}f(x)dx.$$ I know that (and have proved) $$(b-a)\cdot ...
3
votes
1answer
52 views

$\phi(v)/\Phi(v)$ is decreasing for $\phi$ and $\Phi$ being the PDF and CDF of $N(0,1)$

Let $\phi(v)$ and $\Phi(v)$ denote, respectively, the PDF and CDF of the standard normal distribution. How would one show that $$ \frac{\phi(v)}{\Phi(v)} $$ is decreasing? I tried the quotient rule ...
0
votes
2answers
28 views

Bounds on algebraic equation

I have to show that: $$ \frac{-\sqrt{(-\beta -\kappa \sigma -1)^2-4 \beta}+\beta +\kappa \sigma +1}{2 \beta } < 1 $$ I am not sure it is possible. The constraints on the coefficients are: ...
3
votes
3answers
154 views

Problem similar to Kolmogorov's inequality using martingale.

Suppose that $X_k$ is a sequence of independent random variables with mean zero and variance $1$. Let $S_k=X_1+\cdots+X_k$ and let $$ h(\lambda)=\limsup_{n \rightarrow \infty}P\left(\max_{1\leq k\leq ...
2
votes
3answers
215 views

How to find a solution for this inequation?

what's the best way to find a solution for the following inequation: $$ \sqrt{x^2-1}>x $$ The result is as Wolfram says: $$ x \leq-1 $$
9
votes
4answers
673 views

Show that $\frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+…+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+…+a_n$

Showing that $ \frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+...+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+...+a_n$ holds for positive $a_i$s. I've tried adding $a_1+a_2, a_2+a_3,...,a_n+a_1$ respectively ...
-3
votes
4answers
57 views

Show that for real $a,b,c$, $a^2+b^2+c^2\ge ab+bc+ca$

Show that for real $a,b,c$, $a^2+b^2+c^2\ge ab+bc+ca$ We can do this in two obviously trivial ways, that is AM-GM inequality, and its equivalent system, the whole square method. Another way could be ...
1
vote
2answers
144 views

Inequality (absolute value)

$$|x-4|^2 -5|x-4| +6 > 0$$ How can I get rid of the absolute value? Does it work the same way equations with absolute value work?
1
vote
2answers
64 views

For every $n$ there exists $k_n \in \mathbb{N}$ such that $a+k_n/2^n$ is an upper bound while $a+(k_n-1)/2^n$ is not

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
1
vote
1answer
62 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
2
votes
2answers
66 views

Find max of $P=\frac{2}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+1}}+\frac{1}{\sqrt{c^2+1}}$

Give $a,b,c>0$ and $ab+bc+ca=abc$ Find maximum of $$P=\frac{2}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+1}}+\frac{1}{\sqrt{c^2+1}}$$ Could someone help me solve this?
1
vote
2answers
58 views

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number?

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number? I have tried it both with cartesian and polar coordinates but did not get a solution. I got that far: $z=x+yi$ and then I got: $$\pm x ...
2
votes
3answers
121 views

Given $a,b,c$ are the sides of a triangle. Prove that $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}<2$

Given $a,b,c$ are the sides of a triangle. Prove that $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}<2$. My attempt: I could solve it by using the semiperimeter concept. I tried to transform this ...
0
votes
1answer
61 views

Inequalities textbook request [duplicate]

At university I have got a problem set with lots of inequalities. Unfortunately there are no explanations given how to do them. In Highschool we only did very easy inequalities. Therefore I am looking ...
2
votes
1answer
49 views

How to solve this Complex inequality system

$1 \leq z \overline {z} \leq 4$ and $|\Im(z)|<\Re (z)$ How can I solve this system of inequalities? ($\Im$ is the imaginary part and $\Re$ is the real part of a complex number). I have tried so ...
1
vote
1answer
64 views

Let $a_n \rightarrow a$. Show that $\liminf(a_n-b_n)=a-\limsup(b_n)$.

Assignment: Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences of real numbers with $a_n \rightarrow a \in \mathbb{R}$. Show that: $$\liminf(a_n-b_n)=a-\limsup(b_n)$$ ...
0
votes
1answer
53 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...
0
votes
0answers
74 views

Question about Cauchy-inequality

Let $f:\mathbb{C} \to \mathbb{C}$ a holomorphic function and assume that there exist $M > 0$ and $r>0$ such that $$ |f(z)| \leq M |z|\ln |z| $$ $\forall z \in \mathbb{C}$ with $z \geq r$. I ...
4
votes
2answers
179 views

How to proof the following Gronwall type inequality?

Suppose that $g,k: [0,a] \to \mathbb R$ are continuous, $a >0 $, $\,k(t) \ge 0$,$\ c(t) \in C^1([0,a])$, $\, \dot c(t) \ge 0 $ (i.e. $c(t)$ is non decreasing) and $g(t)$ satisfies $$g(t) \le ...
1
vote
2answers
36 views

inequation with complex solutions

Could somebody please help me solve this inequality: $|x-2| < x|x|$ I tried to solve it by using three different values of $x$: 1. $x < 0$ Solution : $1/2 - \sqrt{7}i/2 < x < 1/2 + ...
0
votes
2answers
30 views

Proof of inequality of sums

I have the following to prove, with induction and any help would be appreciated! :) $n\in \mathbb{N}, \quad \left(\, x^{1},\ldots,x^{n}\,\right)\in\mathbb{R}^{n}$ $$ \left(\,\sum^{n}_{i\ =\ ...
2
votes
3answers
91 views

Inequality proof for $1+x^3\geq x+x^2$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ...
0
votes
2answers
25 views

Proof of an inequality in $\mathbb{R}$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. $x,y\in\mathbb{R},\quad \epsilon\in\mathbb{R}:\epsilon\not=0$ $$ ...