The inequality tag has no wiki summary.
0
votes
2answers
15 views
Lower bound on a minimum of maximum of a sequence of standard normal random variables
Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ entries.
We know that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at least $1-\delta$.
I would ...
2
votes
0answers
36 views
Generalizing an approach to proving AMGM
This problem is Exercise 5.5.30 of "The Art and Craft of Problem Solving" by Paul Zeitz.
The problem asks to use the identity
$$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$
to prove the AMGM ...
1
vote
2answers
67 views
An inequality for two complex numbers
I recently saw the following inequality for complex numbers:
If $a,b\in\mathbb C$ and $|a + b|$ and $|a-b|$ are each less than or equal to 1, then
$$|a| + |b^2|/2 \leq 1.$$
How can one prove this?
2
votes
1answer
41 views
Upper bound for $n^{th}$ power of a sum [closed]
Possible Duplicate:
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
We can use Young's inequality to show that $(a+b)^2 \leq 2a^2 + 2b^2$.
Does a similar ...
1
vote
2answers
71 views
How to solve this inequation
Given two real numbers $0<a<1$ and $0<\delta<1$, I want to find a positive integer $i$ (it is better to a smaller $i$) such that
$$\frac{a^i}{i!} \le \delta.$$
1
vote
5answers
95 views
elementary inequality proof
I am working on a howework question, trying to prove the following:
$$5a+b > 4\sqrt{ab},$$
where $a$ and $b$ are positive real numbers.
I've tried multiplying expression by $\sqrt{ab}$, squaring ...
2
votes
1answer
32 views
the scaling problem of Sobolev inequality
How to show that by replacing $\psi(x)$ with $\psi(\lambda x)$, an inequality of the form
$$
\int|\nabla\psi|^2dx\geq C\left(\int|\psi|^qdx\right)^\frac{2}{q}
$$ can only hold for $q=6$ in $N=3$?
0
votes
0answers
16 views
Extending a linear system to satisfy some more constraints
Suppose I have a linear system $Ax = b$ where $x = [x_1, x_2, x_3]$ and it has at least one solution. I also have following constraints on solution $x_1 \ne x_2 \ne x_3 $. Usually we check if any of ...
1
vote
1answer
47 views
An inequality $|x^{1/n}-y^{1/n}|\leq c|x-y|^{1/n}$
I need to prove that $|x^{1/n}-y^{1/n}|\leq c|x-y|^{1/n}$, where $x,y\in [0,+\infty)$, $n\in\mathbb{N}-\{0,1\}$, and $c=2^{(n-1)/n}$.
I tried a lot ways for proving cases $n=2$ and $n=3$, but are ...
2
votes
1answer
49 views
Help with question involving limits, bounds, inequalities
Would someone like to help with the following question?
Prove that for $n=1,2,\ldots$
(a) $5\leq (4^n+5^n)^{1/n}\leq 10$ and that $(4^n+5^n)^{1/n}$ is bounded,
(b) $(4^n+5^n)^{1/n}\geq ...
8
votes
1answer
78 views
Factorial Inequality problem
I met an inequality, I ask, do not mathematical induction to prove that:
Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction
0
votes
0answers
19 views
How to calculate (using a program) a column of sums or to test inequalities? [migrated]
Each expression is on a separate line.
Given
34-2
34-5
34-3
I'm looking to obtain
34-2=28
34-5=29
34-3=31
Or, this would also be helpful:
given:
34-2=5
34-5<=34
34-3=31
I'm looking to ...
5
votes
3answers
104 views
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrasingly weak -
I would like to show that, for a real number $p ...
2
votes
2answers
120 views
How to prove this inequality about $e$? [closed]
Possible Duplicate:
Proving $(1 + 1/n)^{n+1} \gt e$
How to prove this:
$$
\left(\frac{x}{x-1}\right)^x \geq e
\qquad\text{for}\qquad
x \in \mathbb{N}^*
$$
$e$ is the base of the natural ...
4
votes
1answer
72 views
Is my proof for $\sum_{i=1}^{n}x_{i}y_{i}\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}}$ correct?
Is this part of my proof by induction correct ?
$\sum_{i=1}^{n}x_{i}y_{i}\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}}$
this is true when the true is that :
$\sum_{i=1}^{n}\left ...
1
vote
1answer
51 views
Proof by induction of triangle inequality in Hilbert space.
I've made proof by induction over $n$ for triangle inequality : $\left \| x+y \right \|_{e}\leq \left \| x \right \|_{e}+\left \| y \right \|_{e}$
,where $\left \| x \right ...
0
votes
0answers
46 views
Inequality estimation
Let $B$ be an open unit ball in $\mathbb{R}^d$ centered at the origin and $u$ be a twice continuously differentiable function on $\bar{B}$ with $u|_{\partial B} = 0$.
Know
$$\Delta u = f.$$
How can I ...
2
votes
1answer
76 views
Inequalities for integrals
In my notes it was said $$\begin{eqnarray*} n!\int_x^\infty \frac{e^{-y}}{y^{n+1}} \, dy &<& \frac{n!}{x^{n+1}}\int_x^\infty e^{-y} \, dy \\
&=& ...
0
votes
0answers
18 views
Minimal modulus for the finite field NTT
I need your support.
Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity.
I am using it to compute the convolution of two vectors of ...
2
votes
2answers
65 views
a proof for the following Gamma function inequality
Could you please provide or point me to a proof of inequality 5.6.8 found at this site? That is,
$\left|\frac{\Gamma(z+a)}{\Gamma(z+b)}\right| \leq \frac{1}{|z|^{b-a}}$
for $z\in \mathbb{C}$, ...
6
votes
2answers
87 views
Absolute value inequality, where am I wrong?
Tried to solving $|x^2-5x+5|<1$ using the square method, but I don't know what I did wrong:
$$-1<x^2-5x+5<1$$
$$-6<x^2-5x<-4$$
...
9
votes
4answers
296 views
How to know if $\log_78 > \log_89$ without using a calculator?
I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values ...
0
votes
0answers
40 views
Inequality - domain of a variable
For $k>0,x\geq y>0$ the following holds
$$|l(x)| \leq k+|l(y)|,$$
for $x \in (y+k-1,y+k)$.
Now, why does the same inequality hold also for $x \in (y,y+k)$?
Edit: The different posts where ...
1
vote
1answer
37 views
Implication of an inequality
We know that
$$|l(x+u)-l(x)|<1 \text{ for } x\geq y>0 \text{ and } u\in[0,1]$$
Why does:
$$|l(y+u)|<1+|l(y)|,x \in (y+1,y+2)$$
imply that
$$|l(x)| \leq 1 + |l(y+1)| \leq 2+|l(y)|$$
1
vote
3answers
31 views
Inequality absolute value
If $x>0,y>0$, why does it follow for any $y \in (x,x+1)$, that from:
$$|f(y)-f(x)|<1,$$ we have $$|f(y)| \leq |f(x)|+1$$
1
vote
1answer
72 views
Prove by induction that $n^3 > n^2 − 6n + 4$ for all $n ∈ {\mathbb N}$ with $n ≥ 1$ .
Would you please check if my answer is correct and confirm that it is proof by induction? Thank you.
The proof is by induction.
Base Case: when $n=1$:
...
0
votes
2answers
51 views
Solving an inequality modulo 1
In essence, my problem boils down to finding all $i$ that satisfies this inequality ($n$ is constant):
$$
\frac{n}{i} \text{ (mod 1) } < \frac{n}{i+1} \text{ (mod 1) for }n,i\in\mathbb{N}, i < ...
1
vote
1answer
64 views
Greater than zero?
I need to show that
$$\sum_{i=k^*}^K\binom{K}{i}a^{i-1}(1-a)^{K-i-1}(i-aK)>0$$
given $K\geq k^*$, $0<a<1$ and $K$, $k^*\in\mathbb{Z^+/1}$.
I did some computer simulation and saw that it ...
1
vote
2answers
63 views
How do I solve this inequation system
I have the system of equations
$$\begin{eqnarray} 3a &=& A \\ 4b &=& B \\ 7c&=& C \\ a+b+c&=&S \end{eqnarray}$$
subject to
$$A>S,\quad B>S,\quad C>S$$
How can ...
-1
votes
1answer
87 views
Prove or give a counter-example for the inequality
Prove or give a counter-example for the following:
$\frac{2}{\gamma}[\sqrt{(1+\gamma (n-1))(1+\gamma (s -1))}-(1+\gamma (s -1))] \leq n-s$
where $n,s$ natural numbers with $n \geq 2$, ...
0
votes
0answers
53 views
Inequality with convex combination
Consider vectors $v_i \in \mathbb{R}^n$, $z_i \in \mathbb{R}^m$, $i = 1,2,\ldots,N$, and matrices $X$ (positive definite), $F$, $G$ (of appropriate dimensions).
Consider $\alpha_i \in ...
4
votes
2answers
90 views
Proving inequality with powers
Prove that $x^{16}-x^{11}+x^6-x+1>0$ for $x\in R$.
So I thought of something like this:
$$x^{10}(x^6-x)+x^6-x>-1$$
$$(x^{10}+1)(x^6-x)>-1$$
But it seems to not be too much of help. While ...
6
votes
3answers
156 views
How to prove $\cosh(x) \ge 1$ without the $\cosh^2x-\sinh^2x=1$ identity
I think it's pretty easy question, maybe even dumb one, but still I can't find a nice way to solve it.
How do you prove that $\forall x . \cosh(x) \ge 1$ , without using the identity: ...
2
votes
3answers
107 views
Advice on using the Karamata Inequality
Is it possible to use the Karamata inequality to prove the following inequalities to be true ?
$$x^{4}+y^{4}+z^{4}\geq x^{2}yz + xy^{2}z + xyz^{2}$$
$$x^{5}+y^{5}\geq x^{3}y^{2} + x^{2}y^{3}$$
...
1
vote
0answers
32 views
Find the nearest point that lies within a polygon specified by 6 linear inequalities
I have six linear inequalities that together specify a polygon. For a given point $P$, how can I find the nearest point $P'$ that satisfies all six inequalities (if $P$ itself does not)?
[edit: ...
1
vote
2answers
83 views
Inequality holds?
Can anyone prove that
$$
\frac{\sum\limits_{i=1}^{k*} a_i i (x+\epsilon)^{(i-1)}}{\sum\limits_{i=1}^{k*} a_i (x+\epsilon)^{i}+k^*-1}>\frac{\sum\limits_{i=1}^{k*} a_i i ...
1
vote
1answer
41 views
How do I go from a summation form to integral form of an expression?
How is the following inequality valid?
$$\sum\limits_{j = 1}^{N/2} \sum\limits_{n = j}^{\infty} \frac{1}{n^2} < \sum_{j = 1}^{N/2} \int_{j - 1/2}^{\infty} \frac{dx}{x^2}$$.
I came across this ...
1
vote
10answers
309 views
How to prove that for $x$,$y$ positive if $x > y$, then $\frac{1+2x}{1+x} > \frac{1+2y}{1+y}$?
How can I prove this inequality?
$$\frac{{1+2x}}{1+x}\geq \frac{1+2y}{1+y}$$
3
votes
2answers
58 views
Norm inequality on Complex Numbers.
For $z,w \in \mathbb{C}$, it is true that $ 2 | z w| \leq |z|^2 + |w|^2 $. How does this imply the identity: $$|z+w|^2 \leq 2(|z|^2 + |w|^2 )? $$
2
votes
1answer
68 views
Convex hull of sets defined by (in)equalities
If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
4
votes
4answers
143 views
Proving $|x(t)|\lt |t|$ for the solutions $x$ of an ODE
The problem is:
Let $x(t)$ a solution of $$x'=-x(t^2-x^2),$$ so that $|x(t_0)|\lt |t_0|$. Show that for all $|t|\gt |t_0|$, $|x(t)|\lt |t|$.
Suppose that $t_0\geq 0$. Consider
Then $(t_0,x(t_0))$ ...
1
vote
3answers
89 views
Triangle Inequality Proof
I need this one result to do a problem correctly.
I want to show that for any $b \in \mathbb{C}$ and $z$ a complex variable:
$$ |z^2 + b^2| \geq |z|^{2} - |b|^{2}$$
My attempts have only led me to ...
4
votes
1answer
61 views
inequalities and zeroes of a function
Let $f:[a,b]\to\mathbb{R}$ be a smooth function. I am looking for a list of non-trivial inequalities that hold if $f$ has at least one zero in its domain. For example Wirtinger inequality.
2
votes
1answer
74 views
inequality proof of $x^{y-1} \ge xy$
How to prove $x^{y-1}\geq xy$ with
$x,y\in \mathbb{R}$ with $x,y\geq 3$ . Do I need induction? Or is there an elegant way?
4
votes
2answers
78 views
Stuck on simple proving
Prove that $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots-\frac{1}{2009}+\frac{1}{2010}<\frac{3}{8}$
Oh my, I feel embarrased for not knowing how to solve such an elementary ...
1
vote
3answers
46 views
Does $|x|^p$ with $0<p<1$ satisfy the triangular inequality on $\mathbb{R}$?
I am curious about whether $|x|^p$ with $0<p<1$ satisfy $|x+y|^p\leq|x|^p+|y|^p$ for $x,y\in\mathbb{R}$.
So far my trials show that this seems to be right... So can anybody confirm whether this ...
2
votes
3answers
112 views
Combinatorial inequality $\binom{n}{j}\leqslant 2^n$
I was trying to prove (or to find a counterexample) of the following inequality:
$$\binom{n}{j}\leqslant 2^n$$
As I coudn't find a proof/counterexample, I tested some numbers and could see it ...
3
votes
1answer
54 views
inequality with modulus of complex number
Let $ \displaystyle{ z_1, z_2 \in \mathbb{C} }$ where $ z_1, z_2 \neq 0$
Prove that: $\displaystyle |z_1 +z_2| \geq \frac{1}{2} \left( |z_1|+|z_2| \right) \left|\frac{z_1}{|z_1|} + ...
0
votes
2answers
74 views
Why use absolute value for Cauchy Schwarz Inequality?
I see the Cauchy-Schwarz Inequality written as follows
$$|\langle u,v\rangle| \leq \lVert u\rVert \cdot\lVert v\rVert.$$
Why the is the absolute value of $\langle u,v\rangle$ specified? Surely it ...
0
votes
1answer
41 views
Establishing an inequality.
Let $f,x^2f \in L^2(\mathbf R)$. How can I show that $f\in L^1(\mathbf R)$ and
$$ \|f\|_1 \leq \sqrt 2 \|f\|_2 + \frac{\sqrt 6}{3} \|x^2 f\|_2?$$ I have no idea where to begin.
