Questions on proving and manipulating inequalities.

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11
votes
5answers
141 views

How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
5
votes
2answers
189 views

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let ...
2
votes
2answers
39 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 ...
0
votes
1answer
26 views

Consider $f(x) = \frac{2x^3-1+\sin x}{x^2-3}$. Show that $f (x) < 2x$ for most negative values of $x$.

Consider $$f(x) = \frac{2x^3-1+\sin x}{x^2-3}$$ Show that $f (x) < 2x$ for most negative values of $x$. How do I start this/ what concepts does this questions test?
2
votes
0answers
25 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
1
vote
1answer
34 views

Sum of trigonometric functions

Is the following inequality true? $$\left|\sum_{i=1}^{n}\left(\cos(x_i) \prod_{j\neq i}\sin(x_j)\right)\right|\le 1$$ I tried to count the extremes but it didn't work.
4
votes
1answer
57 views
+50

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 ...
0
votes
1answer
30 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. ...
0
votes
2answers
23 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 ...
1
vote
0answers
21 views

Finding maximum value of a 3-variable function using inequality. [on hold]

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
2
votes
2answers
22 views

Inequalities with arctan

I don't understand how to solve inequalities with arctan, such as: $$\arctan\left(\frac{1}{x^2-1}\right)\ge \frac{\pi}{4} $$ If someone could solve this and give me a very brief explanation of what ...
0
votes
1answer
15 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
votes
2answers
23 views

Solving Inequalities with the use of their properties and cases [on hold]

Solve following inequality $$\dfrac4x + 3 \gt \dfrac2x + 1$$ and then graph the solution set on real number line.
0
votes
2answers
76 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 ...
2
votes
1answer
520 views

How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$ [duplicate]

The inequalities are: $$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
2
votes
1answer
38 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
1answer
59 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 ...
2
votes
2answers
35 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 ...
1
vote
0answers
30 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 $n$ ...
1
vote
3answers
45 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?
6
votes
4answers
346 views

Proof of Inequality using AM-GM

I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem: If $a, b, ...
1
vote
1answer
35 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 ...
2
votes
6answers
117 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. ...
3
votes
2answers
34 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
0answers
13 views

Prove $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$ for $F$ is the standard normal CDF

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 one ...
4
votes
1answer
336 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
1
vote
1answer
45 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 ...
1
vote
0answers
33 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, ...
1
vote
1answer
19 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
0answers
54 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
1answer
34 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
15 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} + ...
2
votes
1answer
32 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 ...
2
votes
2answers
242 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ...
1
vote
1answer
25 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 ...
-4
votes
0answers
23 views

Euclid geometry-triangle inequality [on hold]

How to prove that the sum of the diagonals of a convex pentagon are larger than the scope of the pentagon?(using that a+b>c in triangle?
9
votes
4answers
620 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 ...
2
votes
6answers
221 views

Prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c}$ for $a\gt0$, $b\gt0$, $c\gt0$

Please help me for prove this inequality $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c}$, for $a>0$, $b>0$, $c>0$.
-1
votes
3answers
95 views

Prove that this inequality $5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$

Let $a,b,c>0$ and $a+b+c=1$ Prove that $$5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$$
1
vote
2answers
39 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
3answers
96 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 ...
2
votes
4answers
60 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
4answers
518 views

Show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{9}{a+b+c}$ for positive $a,b,c$ [duplicate]

Show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{9}{a+b+c}$, if $a,b,c$ are positive. Well, I got that $bc(a+b+c)+ac(a+b+c)+ab(a+b+c)\geq9abc$.
2
votes
2answers
25 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
2answers
50 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 : ...
0
votes
0answers
33 views

Inequality involving factorial and a number 1/12

How I can prove the following two inequalities: If $n$ is a positive integer then $$ \sqrt{2 \pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12n+1}}<n!<\sqrt{2 \pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12n}} $$
1
vote
5answers
108 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 $\bigg(1+\dfrac 1x\bigg)\bigg(1+\dfrac 1y \bigg)\bigg(1+\dfrac 1z \bigg)\ge 64$ ?
-1
votes
1answer
34 views

Trick with numbers, sums, cubes, squares? [on hold]

Let $(X_1, X_2, \ldots , X_n) = 10$ be a sum of positive numbers where $(X_1^2, X_2^2, \ldots , X_n^2) \geq 20$ is a sum of their squares. Prove that $(X_1^3, X_2^3, \ldots , X_n^3) \geq 40$.
0
votes
1answer
40 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 ...
1
vote
1answer
28 views

Inequalities - proof by induction

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, ...