Questions tagged [functional-inequalities]
For questions about proving and manipulating functional inequalities.
388
questions
1
vote
1answer
35 views
Doubt obtaining the inequality (3) via (1)
I am reading a proof from the 2009-paper Travelling waves for the Gross-Pitevskii equation II (Béthuel, Gravejat, Saut) and I am really stuck in one step. Please, help me with this:
In the paper, ...
2
votes
0answers
73 views
Growth of the gradient of $f(x+y) \leq f(x) f(y)$
Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}_{\geq 0}$ be a radial continuous function and $C^2$ on $\mathbb{R}^3 \setminus \{0\}$ which satisfies the following functional inequality
$$ f(x+y) \leq f(...
1
vote
1answer
112 views
Show that $\chi^2(\nu κ,μ)\le c\chi^2(\nu,μ)$ for all probability measures $\nu$ implies $\text{Var}_μ[κf]\le c\text{Var}_μ[f]$ for all $f\in L^2(μ)$?
Let $(E,\mathcal E)$ be a measurable space and $$\chi^2(\nu,\mu):=\begin{cases}\displaystyle\mu\left|\frac{{\rm d}\nu}{{\rm d}\mu}-1\right|^2=\mu\left|\frac{{\rm d}\nu}{{\rm d}\mu}\right|^2-1&\...
2
votes
0answers
25 views
Can we infer convergence in total variation distance from a Poincaré inequality?
Let $(E,\mathcal E,\mu)$ be a probability space, $\lambda>0$ and $\kappa_t$ be a Markov kernel on $(E,\mathcal E)$ with$^1$ $$\operatorname{Var}_\mu\left[\kappa_tf\right]\le\operatorname{Var}_\mu\...
0
votes
1answer
26 views
Sufficient conditions for the derivative of a quotient to be positive
Suppose we have $g:(0,a)\to(0,b)$ and $h:(0,a)\to(0,c)$ for some positive constants $a$, $b$, and $c$. Now let
$$
f(x)=\frac{g(x)}{h(x)}.
$$
I am trying to determine if
$$
\begin{aligned}
&0<h^\...
2
votes
2answers
56 views
Find function with inequality for derivative
Let $C\geq 2$ and $L>0$. Does there exist $g \in C^1([0,L])$ such that
\begin{equation*}
g(x)>0, \qquad g'(x)>0, \qquad g'(x) > (g(L) - g(x))C
\end{equation*}
holds for any $x \in [0,L]...
2
votes
1answer
35 views
How to Calculate the norm of a matrix transform from $(R^n,\|\cdot\|_{\infty})$ to $(R^n,\|\cdot\|_{1})$
suppose we have a matrix operator $A=(a_{ij})_{n\times n}:(R^n,\|\cdot\|_{\infty})\to(R^n,\|\cdot\|_{1})$, how to calculate $\|A\|$? or if there really is an exactly closed form for $\|A\|$?
if $x\in ...
2
votes
2answers
36 views
How does $\| \nabla u \|_{L^2}^2 \leq C \|\nabla u \|_{L^2}$ imply $\| u\|_{W^{1,2}}^2 \leq \| u\|_{W^{1,2}}$?
So I'm working on some notes and I found this inequality that I really can't make sense of it. This is what is going on:
Let $u \in W_0^{1,2}$ and let $f \in L^2$ both on some $\Omega \subseteq \...
1
vote
1answer
32 views
Automating the solution of pairs of polynomial inequalities as a bound
Let's consider the following pair of quadratic inequalities:
$$\begin{aligned} x^2+x &\geq a\\ x^2-x &< a\end{aligned}$$
The solution of the first inequality is
$$\left( x \geq \frac{-1+...
2
votes
0answers
24 views
Does there exist any solution for this inequality?
Let $\Omega,\Omega^*$ be disks in $\mathbb{R}^2$, such that $\Omega^*\subsetneq\Omega$ and their boundaries meet at one point (so they are tangent at that point; consider $N((1,0),1)$ and $N((2,0),2)$ ...
0
votes
0answers
11 views
The equivalence of a problem formulated as a variational inequalities and problem stated as system of inequalities and equalities
I'm working through a book. The equivalence of $1$ and $2$ are clear to me. I'm struggling with $3$.
Let $T : L^2(\Omega) \rightarrow L^2(\Omega)$ be a coercive monotone operator, where $\omega \...
2
votes
1answer
34 views
How is this a bound: $|(\iota(Tx))(\varphi)| = |\varphi(Tx)| \leq \Vert \varphi \circ T\Vert \cdot \Vert x \Vert \leq \Vert \varphi \circ T \Vert$
I'm wondering how this normed composition---from this answer---works as a bound. Please read question too.
$$|(\iota(Tx))(\varphi)| = |\varphi(Tx)| \leq \Vert \varphi \circ T\Vert \cdot \Vert x \...
0
votes
1answer
60 views
Obtaining orthogonality from a variational inequality in $L^2$
I'm working on the following problem:
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and let $H$ be a closed linear subspace of $L^2(\Omega)$. Let $\gamma : \mathbb R \to \mathbb R$ be a ...
0
votes
0answers
31 views
Show that $f'(0) \geq -\sqrt2$ [duplicate]
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \geq 0$,$f'(x) \leq 0$ and $f''(x) \leq f(x)$, for all $x \geq 0$. Show ...
1
vote
1answer
57 views
Proving a sequence is Cauchy in metric
Consider the sequence, $f_n(x)= \begin{cases}
(2x)^n & 0 \leq x\leq \frac{1}{2} \\
1 & \frac{1}{2} \leq x \leq 1\\
\end{cases}$
Then we need to show that $\{ f_n\}$ is Cauchy ...
0
votes
0answers
12 views
Notation Question Regarding Integral Kernels
I'm reading a proof of an estimate on an integral kernel. They sponateously introduce notation and I'm having trouble following what's happening. (The offending reference is the bottom of pg 60, Lemma ...
0
votes
0answers
44 views
Inequalities involving the greatest common divisor
Question
What are some inequalities involving the greatest common divisor (GCD) function?
Typical Answer
$$\gcd(a,bc) \leq \gcd(a,b)\gcd(a,c)$$
My Attempt
I tried checking the Wikipedia page, ...
0
votes
1answer
28 views
On the existence of a function satisfying a certain inequality
Let $D$ be an open disk in $\mathbb{R}^n$ ($n\ge 1$). I am wondering if there is a nonnegative function $f:D\rightarrow \mathbb{R}$ with $$f\in H^1(D)\cap L^\infty(D)$$ satisfying $$|\nabla f(x)|\le ...
1
vote
0answers
18 views
Product of distributions satisfying log-sobolev inequality
Let $f,g\in C^\infty(\mathbb{R})$ be two smooth positive functions satisfying $\int f = \int g = 1$. Suppose that both $f$ and $g$ satisfy the log-Sobolev inequality (LSI) with constant $C$, so that
...
0
votes
1answer
32 views
Norm Inequality for 1 Dimensional Sobolev Space
Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in H_{0}^{1}(\Omega)$. By Sobolev Embedding Theorem for 1 dimensional space, we can obtain
$$S ||u||_{p}^{2} \leq ||u||_{H^{1}_{0}(\Omega)...
6
votes
0answers
57 views
Inequality for convex function $f:(0,\infty)\to\mathbb{R}$ [duplicate]
I have been working through the Exercises at the end of Chapter $1$ of Bollobas' Linear Analysis. Chapter $1$ is on inequalities, and the text is fairly brief. I have found the problems unexpectedly ...
-1
votes
2answers
118 views
Let $P$ be a polynomial with positive real coefficients. Prove that if $P(1/x) \geq 1/P(x)$ holds for $x = 1$, then it holds for every $x > 0$.
Let $P$ be a polynomial with positive real coefficients. Prove that if
$$
P\left( \frac{1}{x} \right) \geq \frac{1}{P(x)}
$$
holds for $x = 1$, then it holds for every $x > 0$.
What I did:
I was ...
2
votes
1answer
42 views
Functional Space Inequality for Sobolev Space and Lp Space
Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
4
votes
1answer
82 views
Find $f$ if $f(x)\leq x$ and $f(x+y)\leq f(x)+f(y)$ for all $x,~y\in \mathbb{R}.$
Find the formula of function $f:\mathbb{R}\to \mathbb{R}$ if: $$f(x)\leq x$$
and $$f(x+y)\leq f(x)+f(y)$$ for all $x,~y\in \mathbb{R}.$
Attempt. Identity function $I(x)=x$ satisfies the needed ...
1
vote
1answer
44 views
Bound for $e^{-\alpha x}$
For a part of my proof I need to establish that $e^{-\alpha x} \lt h(x)$, where $\alpha,x \gt 0, $ and $x,\alpha \in\mathbb{R}$. I thought for a while and couldn't find a function independent of $\...
1
vote
1answer
33 views
Alternative form for Liapunov inequality
Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then
$$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$
My teacher ...
1
vote
0answers
17 views
Finding discrete solutions to inequality involving Exponential Integral
I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that
$$-c \text{Ei}\left(-e^{\frac{a-d}{c}} (n+1)\right)+a-...
1
vote
0answers
41 views
How to find which of two events -drawn from a normal distribution- is more likely?
I know the probability of event A is given by:
$$\Phi(f(x)+g(y)) - \Phi(f(x)-g(y)), $$
and the probability of event B is
$$\Phi(m(y)+n(x)) - \Phi(m(y)-n(x)).$$
where $\Phi$ is the cumulative ...
2
votes
2answers
73 views
Inequality for $\sin(20°)$
Prove that $$\frac{1}{3} < \sin{20°} < \frac{7}{20}$$
Attempt
$$\sin60°=3\sin20°-4\sin^{3}(20°)$$
Taking $\sin20°$=x
I got the the equation as
$$8x^3-6x+\sqrt{3} =0$$
But from here I am not ...
4
votes
1answer
40 views
Understanding the proof of an inequality
Basically the method applied is the following, we fix $a=\frac{a_1+a_2+...+a_n}{n}$, if: $$f(x)\ge f(a)+f'(a)(x-a) $$This inequality holds for all x, then summing up the inequality will give us the ...
1
vote
1answer
43 views
Injectivity of integral operators
Let $K:L^2[0,1]^{d_1}\to L^2[0,1]^{d_2}$ be integral operator
$$(Kf)(y) = \int f(x)k(x,y)d x.$$
If $d_1>d_2$ is it possible for $K$ to be injective?, e.g. let's take $d_1=2,d_2=1$.
More generally, ...
0
votes
0answers
14 views
Estimate of a general integral involving laplacians
I have two real functions $u,\eta$ defined on $R^n$ and with compact support so we can do all integrations by parts we want, and we need to estimate $\int | \nabla \eta \cdot \nabla u|^2$ by $\int | \...
5
votes
2answers
75 views
Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.
Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$
I don't know where to start as in is there a ...
-1
votes
1answer
79 views
Matries inequality with norms
Let $P$ and $C \neq0$ a $q \times q$ matrices. I want to prove that there exists a positive constants $\alpha$ such under some assumptions under $P$ we have the inequality
$${\left\| {P\left( {I - C} ...
0
votes
0answers
56 views
Functional is weak lower semicontinuous but not weak continuous
I want to show that the functional
$L(u)=\int_0^1 \sqrt{1+(u'(x))^2} dx$ is lower semicontinuous in terms of weak convergence in $W^{1,p}(0,1), p\in(1,\infty)$ but not continuous. Our definition of ...
1
vote
2answers
45 views
Showing that $f(x)$ is convex in $(0,3)$
I've got the following function: $$f(x)=\frac{1}{16x}-\frac{1}{(x+3)^2} $$
And I wish to show that it is convex in the open interval $(0,3)$, took the second derivative, i.e.$$f''(x)=\frac{1}{8x^3}-\...
1
vote
1answer
55 views
A definite integral inequality
Suppose $f(x)$ has continuous derivative on $[-\pi, \pi]$, $\,f(-\pi)=f(\pi)\,$ and $\,\int_{-\pi}^{\pi}\, f(x)\, dx=0$. Then prove that:
$$
\int_{-\pi}^{\pi} [\,f'(x)]^2\, dx \ge \int_{-\pi}^{\pi} f^...
1
vote
0answers
26 views
Inequality involving Weibull distribution
define R(x) as the weibull survival function
$R_1(t)=e^{-\alpha_1 t^{\beta_1}}$
$R_2(t)=e^{-\alpha_2 t^{\beta_2}}$
with $\alpha_1, \alpha_2 > 0 $ and $\beta_1, \beta_2 >1 $
$\phi_1(t)=\int_{...
0
votes
0answers
51 views
Remark 16 in Chapter 2 from Brezis - Functional Analysis, Sobolev Spaces, and Partial Differential Equations.
Let $E,F$ be two Banach Spaces and $A : D(A) \subset E \to F$ be a linear unbounded operator which is densely defined. Now, we would like to define $A^{*}$ as the adjoint of $A$. Let $A^{*} : D(A^{*}) ...
1
vote
1answer
19 views
Proving that $|x^*(x)| \leq \rho(x,Y)$
Exercise :
Let $(X,\|\cdot\|)$ be a normed space, $Y$ a subspace of $X$ and $x^* \in X$ with $\|x^*\| \leq 1$ such that $x^*|_Y = 0$. Show that $\forall x \in X \setminus Y$, it is : $|x^*(x)| \leq ...
0
votes
0answers
22 views
Proving complicated transcendental inequality
Suppose we have a function $f$ of four posirive real numbers $a,b,c$ and $d$ in a domain that, for a given real number $0<r<1$ they satisfy
$$rc<b<a,$$
$$rc<rd<a.$$
We then have
$$...
0
votes
1answer
28 views
Inequality of derivatives
I'm developing a mathematical model for a physical system and have come across the following logical quandary (at least, it is for me). I'm having trouble proving whether the following proposition is ...
0
votes
0answers
31 views
Jensen type inequality for a non-convex function.
I suppose that a function $f$ is $\geq 0$ on $[-1,1]$, decreasing and $f(t)(1+t)$ is concave.
Moreover for every $a,b \in [-1,1]$, $a<b$, we have a (simple positive) measure $\mu_{a,b}$ such that
$...
1
vote
0answers
36 views
Prove upper bound inequality for the dimension of the space of symmetric tensors
I want to check that the dimension of the space of symmetric tensors $N(n,m) := dim(Sym^m(\mathbb{R}^n))$ satisfies $N(n,m) \leq \frac{n^m}{m!}(1+\frac{2m^2}{n})$.
Thus I need prove inequality.
If $...
2
votes
1answer
63 views
Poincare type inequalities
I want to prove if following inequality holds:
$$\int_0^1(f')^2\ dx\geq f^2(1)-f^2(0)$$
where $f$ is a function in $H^1([0,1])$ satisfying $\int_0^1f \ dx=0$. It is actually a one dimensional case of ...
1
vote
1answer
127 views
Does the following inequality always hold true?
$$ 0\lt \frac{\sum_{i=n}^{n + P_n - 1} P_i}{P_n \cdot P _{P_n}} \leq 1 $$
Or is there a lower bound bigger than zero? Which I believe not to be the case.
Some basic examples are as follows:
$(1)$
...
2
votes
1answer
66 views
RMO practice problem inequality
Let $a_n$ & $b_n$ be two sequences such that $a_0$ , $b_0$ > 0 and $a_{n+1}$ = $a_n$ + $\frac{1}{2b_n}$ & $b_{n+1}$ = $b_n$ + $\frac{1}{2a_n}$ $\forall$ n $\geq$ 0. Then prove that $$max(a_{...
1
vote
1answer
26 views
Equivalence of norm in Banach Space
Let $A$ be a densely defined closed linear operator in a Banach space $X$ and $\sigma(A)$ be its spectrum. We define its spectral radius $r_{A} := \sup\limits_{\lambda \in \sigma(A)}|\lambda|$. Now, ...
0
votes
3answers
36 views
Determine the set of all complex number z satisfying following conditions
I’m having some troubles of calculating complex numbers where I need to deal with absolute values and inequalities.
Here is an example I’ve been working on but I get stuck
Re(2/z)+Im(4/z)<1
I use ...
3
votes
5answers
140 views
Why does proving inductively $n < 2^n$ for $n \geq 1$ imply it is true for real values of $n$?
If we prove by induction that $2^n > n$ for $n \geq 1$ where $n \in N^+$, how can one know this inequality holds for real values of n like $2^{2.5} > 2.5$?
Maybe a bit silly question but I can'...
