1
vote
2answers
21 views

Show that $0\le 1- \frac{n!}{(n-k)!n^k}\le1$

I would like to prove that $$ 0\le 1- \frac{n!}{(n-k)!n^k}\le1 $$ for $n\ge k \in \mathbb{N}$ I tried to use the fact that $n^k\ge n$ and binomial coefficient but it doesn't look good, plus I have ...
4
votes
5answers
109 views

Prove inequality $(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$

I am trying to prove the following inequality: $$(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$$ for all $a>0, b>0$. Does anyone know how to prove it? Thanks a lot in advance! ...
2
votes
3answers
127 views

Integral Inequality

There is a problem in my book that says: Show that (assuming $f,g,f^2, g^2, $and$ fg$ are integrable) $$\int_{a}^{b}fg\leq \sqrt{\int_{a}^{b}f^2}\sqrt{\int_{a}^{b}g^2}$$. I know this is the ...
0
votes
3answers
29 views

Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
7
votes
2answers
86 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
0
votes
1answer
27 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
6
votes
4answers
79 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
0
votes
1answer
32 views

Inequality, $\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$

How do I show that $$\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$$ for $x\in\left(0,1\right]$ and $n\in\mathbb N$?
7
votes
1answer
88 views

How to learn inequalities and become good at proving them?

I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me. The class is quite challenging and the fail rate is relatively high. ...
0
votes
1answer
17 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
0
votes
1answer
15 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
0
votes
1answer
20 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
0
votes
1answer
23 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
4
votes
3answers
50 views

Help verifying my proof that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$

I'm trying to prove that that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$. Obviously $j,k\in \mathbb{N}$. This is not for homework, it's a ...
4
votes
2answers
71 views

Show that $f(x_0)-f(x)<\vert x_0-x\vert$ for all $x \ne x_0$

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ continuous and bounded below. Show that there exist $x_0 \in \mathbb{R}$ such that $\forall x\ne x_0$, $$f(x_0)-f(x)<\vert x_0-x\vert$$ Since $f$ ...
4
votes
3answers
59 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
3
votes
2answers
74 views

$\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$ $\Rightarrow$ $\left|f(x)\right|\le \sqrt3$

Let $f:[0, 1]\rightarrow \mathbb{R}$ be a function that is continous on $[0,1]$ and derivable on $(0, 1)$. If $\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$, show that $\left|f(x)\right|\le ...
1
vote
4answers
147 views

Proving a logarithm inequality

$$\frac{1}{n+1}< \log(1+ 1/n)$$ Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed.
2
votes
1answer
73 views

Prove a real valued function is increasing

Prove $$-\frac{x(1-x^c)\ln x}{(1-x)^2}$$ $c>0$ is an increasing function on $[0,1]$. I have a relatively cumbersome proof sketched below. I would like to see other ideas, particularly simpler ...
0
votes
0answers
31 views

Bounds of the solution space

I have a continuous function $f(x)=a x^2+b x+c$ that is defined on $]0,X[$. I know that the function values are bounded, $Fu <f(x) <Fl$ for all values of $x$. I want to find the bounds on the ...
0
votes
1answer
45 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
0
votes
2answers
37 views

How do I conclude that $G(a)= a_1^\frac{1}{n},…,a_n^\frac{1}{n}$ must obtain its maximum when $a_1=…=a_n = U(a)$?

In the AM-GM Inequality, how do I conclude that $G(a)= a_1^\frac{1}{n},...,a_n^\frac{1}{n}$ must obtain its maximum when $a_1=...=a_n = U(a)$ and ($U(a)= \frac{a_1+...a_n}{n}$ is the arithmetic mean)? ...
1
vote
2answers
91 views

How do I proove $\frac{-1}{1+t^2} e^{-At} (t \sin(A) + \cos(A)) \leq e^{-At}$?

Is there an easy way to proove that $$\frac{-1}{1+t^2} e^{-At} (t \sin(A) + \cos(A)) \leq e^{-At}$$ is satisfied for all $A,t \in \mathbb{R}_{+}$? I need this to understand the pattern solution ...
1
vote
1answer
32 views

Simple inequality in euclidean $n$ space [duplicate]

Let $x_1,...,x_n \in \mathbb{R}$. Then $$ |x_1| + ... + |x_n| \leq \sqrt{n} \cdot \sqrt{ x_1^2 + ... + x_n^2} $$ Is this inequality true? I have proved it for $n = 2$. For instance, it follows ...
1
vote
1answer
44 views

Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
0
votes
1answer
35 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
0
votes
2answers
82 views

Show that if $\displaystyle\int_0^1f(x)dx=a$, then $\displaystyle\int_0^1\sqrt{f(x)}dx\ge a^{2/3}$

$f$ is continuous on $[0,1]$ and there is $a>0$ such that, $0\le f(x)\le a^{2/3}$ for $x\in[0,1]$. Show that if $\displaystyle\int_0^1f(x)dx=a$, then $\displaystyle\int_0^1\sqrt{f(x)}dx\ge ...
1
vote
0answers
46 views

Why doesn't this contradict “weak inequality doesn't imply strict inequality” ?

I question a new from my other question because this issue didn't occur until presently. I understand $0<|x-c|<\delta \implies |f(x)-f(c)|<\epsilon $ and $x = c \implies |f(c)-f(c)| = 0 ...
1
vote
2answers
53 views

Is this function inequality true?

Let $\lambda$ and $\lambda_L$ be the values of the function $f(x,y)$ at the optimum for problems \begin{align} \lambda=\max_{x}\min_{y}f(x,y) \end{align} \begin{align} ...
1
vote
4answers
104 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
0
votes
0answers
25 views

Converge of Sum divide by log(n)

I am trying to show that If $b_n = \sum^n_{k=1}(k^{-1}) -\sum^n_{k=1}(k^{-2})$ then $\frac{b_n}{\log(n)} \rightarrow 1$ as $n \rightarrow \infty$ I start this problem by showing this inequality ...
2
votes
1answer
52 views

Below bound of the mesure of a finite intersection

Let $(X, \mathcal{M}, \mu)$ be a measure space, with $\mu(X)=1$. If $A_{1}, A_{2}, ..., A_{n} \in \mathcal{M}$, prove that $$\mu \left(\bigcap_{j=1}^{n} A_{j} \right) \geq \sum_{j=1}^{n} \mu{(A_{j})} ...
3
votes
1answer
92 views

Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
3
votes
3answers
52 views

Is the following inequality true?

Suppose that $\int_{0}^{1}|f(x)|dx<\epsilon.$ Is the following inequality true $$ \frac{1}{|I|}\int_{I}|f(x)|dx\leq \epsilon $$ for any subinterval $I\subset [0,1].$
4
votes
2answers
66 views

Analysis/Inequality question about proving an infinite product greater than 0

This is from David Williams' book Probability using Martingales. I'm self-studying. Question Prove that if $$0\leq p_n < 1 \quad\text{ and }\quad S:=\sum p_n < \infty$$ then $$\prod (1-p_n) ...
3
votes
0answers
17 views

inequality involving lifts of a positive oriented homeomorphism of the circle

Let $\pi: \mathbb R \to S^1$ be the natural projection and let $f:S^1 \to S^1$ be a positive oriented homeomorphism. We say that $F: \mathbb R \to \mathbb R$ is a lift of $f$ is $\pi \circ F = f \circ ...
4
votes
5answers
257 views

Prove that $\forall x>0, \frac {x-1}{\ln(x)} \geq \sqrt{x} $.

This inequality arose in this question Prove that : $|f(b)-f(a)|\geqslant (b-a) \sqrt{f'(a) f'(b)}$ with $(a,b) \in \mathbb{R}^{2}$ : $$\forall x>0, \frac {x-1}{\ln(x)} \geq \sqrt{x} $$ ...
1
vote
1answer
29 views

help with integral inequality

Let $P(R)=e^R\cdot\int_R^{\infty}F(z)e^{-z}dz=\int_0^{\infty}F(R+z)e^{-z}dz$. Is it true that $P(R) \geq 0$ for all $R$ implies $F(z) \geq 0$ for all $z$? In my case, $F(z)$ is a difference of CDF ...
0
votes
3answers
60 views

Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers

Expressed mathematically, the question is to prove the that $\frac{1}{n}$ $\sum_{i=1}^{i=n}{a_i}\leqslant$ $\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$ First of all, what form of Cauchy-Schwarz should I ...
0
votes
1answer
73 views

Cauchy–Schwarz inequality on vector-valued L2 space

Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking ...
5
votes
3answers
71 views

An inequality from derivatives

Let $f:\;\mathbb{R}\to\mathbb{R}$ be differentiable with $f(0)=0$ and $f''(0)$ exists and is positive. I would like to show that there exists $x>0$ such that $f(2x)>2f(x)$. My try: ...
3
votes
4answers
131 views

Integral inequality for continuous function

Let $ f $ be a continuous, real-valued function on $[0, 1] $. Show that $$\int_0^1 \int_0^1 |f (x)+f (y)| dx dy \ge \int_0^1 |f (x)| dx $$ I tried to dissect the square in triangles and use ...
1
vote
1answer
37 views

Euler-Mascheroni constant bound for the Omega constant

The Omega constant $\Omega = \rm{W}_0(1) = 0.56714329\ldots$ is defined as the real root to the equation $x {\rm e}^x = 1$ and corresponds to the value of the Lambert W function for an argument equal ...
5
votes
2answers
89 views

Inequality involving absolute values and square roots

I could use some help with proving this inequality: $$\left|\,x_1\,\right|+\left|\,x_2\,\right|+...+\left|\,x_p\,\right|\leq\sqrt{p}\sqrt{x^2_1+x^2_2+...+x^2_p}$$ for all natural numbers p. Aside ...
0
votes
1answer
26 views

Inequality on Cardinality of a set

Consider the set $\mathcal{X} := \{ x \in \mathbb{R}^{n} \mid \sum_{i=1}^n x_i = 0, \ \sum_{i=1}^{n} x_i^2 = n \}$. Prove that, for all $k \geq 1$, $$ \max_{ x \in \mathcal{X} } \ \text{Card}\left( ...
0
votes
1answer
56 views

Nature of inequalities near origin

What can be said about these statements if point $(x,y)$ is sufficiently close to origin $(0,0)$. $$x+y+4 \sin x \sin y \ge0$$ $$2x^2 + 3y^2 +4 \sin x \sin y\ge 0$$
2
votes
1answer
94 views

Generalized Hölder inequality, the case when equality holds

I know the generalized Hölder inequality sounds like as: Let $1\leq p_1,\ldots,p_n<\infty$ and $p>0$ such that $\frac1p=\frac1{p_1}+\cdots+\frac1{p_n}$. Then, for all measurable functions ...
5
votes
7answers
219 views

Prove that $\left(\frac12(x+y)\right)^2 \le \frac12(x^2 + y^2)$

Prove that $$\left(\frac12(x+y)\right)^2 \leq \frac12(x^2 + y^2)$$ I've gotten that $$\left(\frac12(x+y)\right)^2 \ge 0 $$ but stumped on where to go from here...
1
vote
2answers
41 views

Prove $(x+y)^a\leq x^a+y^a$ if $0<a\leq1$ and $x,y\geq0$

Prove $(x+y)^a\leq x^a+y^a$ if $0<a\leq1$ and $x,y\geq0$ I need to prove this step for a bigger question. It should be quite basic but I just have no idea...
1
vote
1answer
98 views

How to prove $\left(|a+b|^p+|a-b|^p\right)^{1/p}\ge 2^{1/p}\left(a^2+(p-1)b^2\right)^{1/2}$

For real numbers $a, b$ and all $1\le p\le 2$, how to prove $$\left(|a+b|^p+|a-b|^p\right)^{1/p}\ge 2^{1/p}\left(a^2+(p-1)b^2\right)^{1/2}?$$