Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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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
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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(...
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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
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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\...
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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
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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
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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
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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 \...
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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+...
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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 $N((2,0),2)$ ...
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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
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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
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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 ...
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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
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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 ...
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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-...
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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
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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
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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
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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, ...
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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 | \...
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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 ...
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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} ...
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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 ...
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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
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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^...
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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_{...
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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
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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 ...
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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
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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 ...
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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
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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
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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
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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
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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
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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, ...
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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
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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'...