Questions on proving and manipulating inequalities.
7
votes
1answer
139 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
6
votes
1answer
83 views
An Inequality question
I have the following question. I have to find a $\delta>0$ such that for all complex numbers $x,y$ the following holds true - \begin{equation}
\frac{1}{2\pi}\int_0^{2\pi}|x+e^{it}y|\,dt \ge ...
6
votes
1answer
391 views
Farkas Lemma proof
I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm.
From Wikipedia:
Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
5
votes
1answer
366 views
Least-squares left-inverse having smallest Frobenius norm
While trying to prove that the left-inverse of $A$ provided by the least-squares solution to $y=Ax$ has the smallest Frobenius norm, I am stuck at a point which I describe below:
Let $B$ be any ...
4
votes
1answer
51 views
How prove this inequality $e^{|Im(z)|}\le B|\sin{z}|$
Let $z\in C$ with $|z-n\pi|\ge\dfrac{\pi}{4}$ for all $n\in Z$,Then
$$e^{|Im(z)|}\le B|\sin{z}|$$
find the minimun $B$
I have prove $B\ge 4$,But I think is very ugly, can you have nice methods? and ...
4
votes
1answer
67 views
Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space
Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case?
...
4
votes
1answer
81 views
Prove or disprove an inequality with $0 \le a_1 \le a_2 \le \ldots \le a_n$
Let $n \in \mathbb{Z}_+$ be $n \ge 3$ and $0 \le a_1 \le a_2 \le \ldots \le a_n$. Prove or disprove an inequality:
$$\large \sqrt{a_1a_2} + \sqrt{a_2a_3} + \ldots + \sqrt{a_na_1} \ge ...
4
votes
1answer
177 views
Stochastic integral inequality
Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$.
Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that
...
3
votes
1answer
49 views
How to show the following inequality?
How to show following inequality using Stirling approximation?$$\sum_{i=1}^n(\frac{p}{1-p})^n\cdot\frac{1}{(n+i)!(n-i)!} \leq \frac{1-p}{1-2p}$$
Any kind of hint will be appreciated. Thanks in ...
3
votes
1answer
42 views
How to show a basic integral inequality?
The following inequality is quite clear for $R^1$:
$$\int_{B_1}1/|x-y|^\alpha dx\leq\int_{B_1}1/|x|^\alpha dx,\quad\forall y\in B_1,$$
where $B_1$ is the unit ball in $R^1$, i.e., $[-1,1]$ and ...
3
votes
1answer
60 views
Inequality concerning a Holder continuous function composed with a diffeomorphism
I'm trying to fill in the details for the following inequality from a paper, but am thoroughly stumped.
Prelude
Let $f \in C_c^{\gamma}(\mathbb{R}^n)$ for some $\gamma \in (0,1)$ (that is, a ...
0
votes
0answers
50 views
linear equations with inequality constraints
The problem is, given a set of linear equations $Ax=b$ such that the system is under-determined, and a set of linear inequalities $Cx\geq 0$, find a solution for the system. Does anyone know a general ...
0
votes
0answers
62 views
Prove $\|x-y\|\|x+y\|\le\|x\|^2+\|y\|^2$ for all Rn
Prove $\|x-y\|\|x+y\|\le\|x\|^2+\|y\|^2$ for all Rn
I've been struggling with this for a while and haven't figured out a way to do it either geometrically or algebraically.
0
votes
0answers
51 views
proving an inequality by induction
Not sure how to proceed. I'm trying to prove that the following inequality is true. I know that $t_2 = 6$ and $t_3=17$ from the problem statement. The base case is obvious.
$t_{r+1} \leq (r+1) (t_r ...
0
votes
0answers
121 views
Find function $f$ such that $ (x_1 + f(x) )^6 + ( x_2 + f(x)^3 )^2 - x_1^6 - x_2^2 < 0$
Let $x=(x_1,x_2) \in \mathbb{R}^2$. Find a locally-bounded function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that
$$ (x_1 + f(x) )^6 + \frac{1}{4}( x_2 + f(x)^3 )^2 - x_1^6 - \frac{1}{4}x_2^2 ...
0
votes
0answers
73 views
Find the maximum value of $M=\frac{80a^3}{27}+\frac{9b^3}{4}+\frac{abc}{2}$
Let $a,b,c\in \mathbb{R}$ such that:
$$3 \ge a \ge b \ge c >0$$
$$2a+3c \ge abc$$
$$\frac{18c}{a}+\frac{4a}{b}+\frac{3b}{c} \ge 3abc$$
Find the maximum value of ...
0
votes
0answers
62 views
Prove that $x-\mid y\mid\le\left|\frac{x-y}{1-xy}\right|<1$
Let a be a real number, b is a complex number, $a \in (0,1)$ and $|b|<1$
Prove that $$x-\mid y\mid\le\left|\frac{x-y}{1-xy}\right|<1$$
I have solved the left side: ...
0
votes
0answers
69 views
When $(\sum_{i=1}^nk_i < \prod_{i=n}^ni^{k_i}k_i!)$?
Consider $\Omega \subset \mathbb{N}$ a finite subset of $\mathbb{N}$, $\phi: \Omega \rightarrow \mathbb{N}$ an enumeration of $\Omega$ such that $\phi(\omega)=i$ and $|\Omega|=n$,
$$
...
0
votes
0answers
66 views
Need the solution of Problems of Vasc and Arqady
On the Art of Problem Solving web site, there is a PDF file (Problems Proposed by Vasc and Arqady - Edited by Sayan Mukherjee.pdf) containing 100 problems dealing with solving inequalities.
I know it ...
0
votes
0answers
35 views
Extracting a function from set of inequalities
I have set of inequalities in two dimension space which represent relation between $X$ and $Y$. now I want a function whose input is $X$ and output is $Y$. In other words, I want $F$ such that ...
0
votes
0answers
42 views
Some kind of trace inequality
What is the trick, to prove
$\| u\|_{L^2(\Gamma)} \leq k \frac{1}{r}\| u\|_{L^2(\Omega)} + r \| \nabla u\|_{L^2(\Omega)} $ ?
$\Gamma$ is one side of $\Omega:= [0,r] \times [0,r] $.
I tried partial ...
0
votes
0answers
63 views
When inequality for binomial coefficients is true?
I've asked similar question here Inequality for binomial coefficients, but with slightly different assumptions. I am curious what happend if $m, k$ are fixed.
Let $m \leq n, n \leq N$ and $0\leq k ...
0
votes
0answers
28 views
Inequality on Quotient Substution and Cauchy
Let $n>3$ and for positive $x_1,...,x_n$, and $x_1x_2...x_n=1$. Prove that:
$1/(1+x_1+x_1x_2)+...+1/(1+x_n+x_nx_1)>1$
For this inequality I do not see how to prove it using the conditions ...
0
votes
0answers
61 views
sum to integral inequality step in a proof of Kolmogorov
If I have $N$ numbers $x_j$ very very close to $N$th roots of unity. How could I show $$\frac{1}{N} \sum_{j=1}^N \left|\sin(\tfrac{1}{2}(t-x_j))\right|^{-1} > \int_{1/N}^\pi ...
0
votes
0answers
29 views
What can be said about these ratios of third and first derivatives?
Let's say that we have function $u:\mathbb R_0\to \mathbb R$ with $u'(x)>0$, $u''(x)<0$, $u'''(x)>0$, $\lim_{x\to 0} u'(x) = \infty, \lim_{x\to 0}u'(x) = 0$.
Take $x_1 < x_2$. Does ...
0
votes
0answers
274 views
Binary symmetric channel capacity or mutual information inequality
I proved that
I(X,Y) <= 1 - H(p)
to the following way:
How can I prove if I start in that way I(X,Y) = H(X) - H(X|Y), I ...
0
votes
0answers
58 views
inequality with gamma function
Help me please to prove the following inequality
For $x,y>1, x \neq y$.
$$
\frac{1}{\Gamma(x)\Gamma(y)}\leq 2\sqrt{2\pi}\frac{\sqrt{x+y}}{\Gamma(x+y)}.
$$
Thank you.
0
votes
0answers
81 views
Difficult inequality, difficult solution, but how to prove?
Having
$$ \frac{|x-3|}{x} + |x^2-2x+1| + x > 0$$
how can I arrive to the solution:
$$
x < \frac 13 \left( 1-\sqrt[3]{\frac{2}{79-9\sqrt{77}}} - \sqrt[3]{\frac 12 \left(79 - 9\sqrt{77}\right)} ...
0
votes
0answers
78 views
Proving a simple inequality
Can someone show that the inequality bellow holds?
$$ f(n) \leq f(n+1) \ $$
Where
$$ \frac{\sum\limits_{k=1}^n \Lambda(k) {k}/{n}\lceil{n}/{k}\rceil{}\{ n/k \}}{\sum\limits_{k=1}^n \Lambda(k)}=f(n)$$
...
0
votes
0answers
71 views
an inequality like the triangle inequality
Its easy question, but I cannot find the name of the inequality. Please provide me with it.
I am doing the following, let $a$ be $n$-dimentional vector. Let $b_i, i=1, \ldots, n$ be positive numbers. ...
0
votes
0answers
146 views
Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product
I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product.
Definition. Suppose that $\mathscr X$ is a vector space over ...
0
votes
0answers
97 views
Formulation and solution of non-linear optimzation problem with inequality constraints
I'd like to know if the following problem is well formulated and has solutions. I'm very new to the subject of nonlinear optimization with inequality constraints ('teaching myself the Kuhn-Tucker ...
0
votes
0answers
46 views
How to find the region of system of nonlinear inequalities?
Can someone tell me a numerical method to find the feasible region of system of nonlinear inequalities? I found a method based on Newton Method, but that only finds one point in feasible region. I wan ...
0
votes
0answers
74 views
Integral estimation with the Hölder inequality
How could one estimate $\int\limits_{\Omega}|uv^2|$ with using the Hölder inequality?
0
votes
0answers
94 views
application of some inequalities in optimization theory
I am studding the Theory of Optimization. And it turns out that some classical inequalities (especially like Khinchine, or non-commutative Khinchine inequaliry or Kahane's inequality) are 'very key' ...
0
votes
0answers
40 views
understanding of 1-unconditionality
Let $X=(X, \|\cdot\|_X)$ be normed space with $x_1, \ldots, x_m\in X$.
Assume, $\int_{[-1,1]^m}\|\sum_{i=1}^ma_ix_i\|_Xd\mu(a)=1$, where $\mu$ is the Lebesgue measure on $[-1,1]^m$, $a \in [-1,1]^m$.
...
0
votes
0answers
58 views
condition on non-commutative Khinchine inequality
Let $\epsilon=(\epsilon_1,\ldots \epsilon_M)$ be Rademacher sequence. And let $B_j, j=1, \ldots, M$ be complex valued random matrices of the same dimension. Choose $n\in \mathbb{N}$. Then, the ...
0
votes
0answers
150 views
Schwarz inequality and linear dependence
Let $\{a_i\}$ and $\{b_i\}$ be families of complex numbers.
I know that if $\{a_i\}$ and $\{b_i\}$ are linear dependence, then Schwarz inequality becomes equality, but I cannot prove the converse. ...
0
votes
0answers
58 views
proof of one inequality with sums
Please help me to prove the following inequality:
Fix $k, m \in Z_+$ and for $j \in Z_+$ set
\begin{align*}
a_j^{(1)}=a_j=\sum_{i=0}^{\min\{j,k\}}\frac{1}{i!6^i}\frac{(-1)^{j-i}}{(2(j-i)+1)!}
...
0
votes
0answers
41 views
Exponential decay of optimal stopping rule
I'm trying to prove the following:
For any $\lambda,\tau$, probability distribution, if $T$ is an optimal stopping rule from $\lambda$ to $\tau$ then for all $k\geq 1$,
$$
...
0
votes
0answers
52 views
How to prove inequality_3
$\Bigl( \sum\limits_{i=1}^{n-1} a_{i}^{2}\Bigr) \cdot \Bigl( \sum\limits_{j=1}^{n-1} b_{i}^{2}\Bigr)-\Bigl( \sum\limits_{l=1}^{n-1} a_{l}b_{l}\Bigr)^{2}-2\cdot\lceil\frac{n-1}{2} \rceil\cdot \Bigl( ...
0
votes
0answers
68 views
How to prove inequality_2
I try to prove such inequality $\forall n$:
$\Bigl(1+\sum\limits_{l=1}^{n-1} a_{l}^{2}h^2 \Bigr) \cdot \Bigl(1+\sum\limits_{l=1}^{n-1} b_{l}^{2}h^2 \Bigr)-\Bigl(\sum \limits _{l=1}^{n-1} ...
0
votes
0answers
144 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 ...
0
votes
0answers
52 views
Solving $$ b^n +nc+d\leq 0 $$
Can someone give me a hint how to solve the inequality $$ b^n +nc+d\leq 0 $$
for $n\in \mathbb{N}$, where $b,c,d\in \mathbb{R}$ and $-1\leq d\leq 0,\ c\leq -1$ and $c\geq 2$?
I think I need some ...
0
votes
0answers
208 views
Inequality with Stirling's numbers
I supect that for all $n>k>0$:
$k^2\left\{ \begin{array}{c}n\\k\end{array} \right\}^2 +2k\left\{ \begin{array}{c}n\\k\end{array} \right\}\left\{ \begin{array}{c}n\\k-1\end{array} ...
0
votes
0answers
145 views
How to solve these inequations?
$C_i$ is a $k_i\times N$ matrix over finite field $\mathbb{F}_q$, where $i\in \{1,2,\ldots,K\}$, $k_i<N$, and $q<K$. My questions are 1) how to determine whether there is a $1\times N$ vector ...
0
votes
0answers
45 views
Asymptotic planes in rectangular coordinates
I was trying to prove the triangle inequality theorem (the sum of two sides of a triangle are always greater than the third) with simple algebra, and it eventually "boiled down" to proving that ...
0
votes
0answers
140 views
An estimation of $\sqrt[n+1]{x}$ given $\sqrt[n]{x}$?
I have a sequence $x_n$ and I want to prove that $\sqrt[n]{x_n}\le\sqrt[n+1]{x_{n+1}}$ for every $n$. The problem is I don't know how to handle the transition from $n$ to $n+1$ in the exponent. Are ...
0
votes
0answers
92 views
Isoperimetric inequality transformed to an integral inequality
In the following article, it is shown that Gage's inequality (a form of isoperimetric inequality) can be generalised as the following integral inequality which is presented as a conjecture in the ...
-1
votes
0answers
43 views
Prove the inequality $f(m_{1},m)+f(m,m_{2})\ge 1+f(m_{1},m_{2})$
Let $x,y,a_{i}\in R^{n}$ be real numbers,and $0\le m_{1}\le m\le m_{2},0\le a_{i}.i=1,2,\cdots ,k$
and
...
