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Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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29 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
56 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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0answers
38 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
22 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
9 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
21 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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56 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
108 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
39 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 ...
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1answer
75 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
42 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 $\...
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1answer
28 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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16 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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40 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 ...
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2answers
71 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 ...
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1answer
39 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 ...
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30 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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12 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
73 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
51 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
45 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
44 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}-\...
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1answer
48 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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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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41 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^{*}) ...
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1answer
17 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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21 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 $$...
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1answer
25 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
26 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 $...
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0answers
27 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 $...
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1answer
57 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 ...
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1answer
124 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: $(1)$ ...
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1answer
63 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_{...
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1answer
24 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
34 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 ...
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5answers
133 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'...
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1answer
66 views

How is this proposition true?

I have a function of $n$, where $n$ is an integer, defined as follows $$D(n)=\frac{nK}{K+\bigg\lceil\frac{n}{M}\bigg\rceil(n-1)},$$ where $K$ are $M$ are some constant positive integers. For this ...
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0answers
36 views

How to prove the following inequality for a function?

I have function $$D(M\gamma)=\frac{M\gamma K}{K+\gamma(M\gamma-1)}\tag{1}$$ where $K$ is some positive constant. In this case, how to show that $$D(M(\gamma-1))<D(M\gamma),~~\text{for } \gamma<\...
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1answer
21 views

Best way to check to see if conditions are (or can be) satisfied?

Sorry to bother you guys, but a project I'm working on necessitates checking to see whether somewhat obscure conditions can all hold, given some bounds on possible values for each variable of interest....
3
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1answer
112 views

Interpolation inequalities

Let $\Omega$ be a regular domain of $\mathbb{R}^d$, $d=2,3$. Let $\mathcal{T}_h$ be a triangulation of $\Omega$ of size $h>0$. Assume we can prove \begin{equation*} \begin{aligned} \|v\|_{L^2(\...
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0answers
49 views

A generalisation of a Hardy inequality

I found the following result in Brezis' book. Let $\Omega$ be a bounded open set of class $C^1$. Let $d(x):=\operatorname{dist}(x,\partial\Omega)$. Then there exists $C>0$ such that $$ \|\...
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1answer
51 views

Equality of functions on real numbers implies equality on rationals?

Define a pair of functions $p, h: \mathbb{Q} \to [0,1]$ over the rational numbers, with the properties \begin{align} 0 \le p(x) \le 1 && 0 \le h(x) \le 1 && (x\in\mathbb{Q}), \\ \end{...
2
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0answers
84 views

Problem in proving fixed point theorem

Let $ f(x)=x^r, 1<r<\infty, x\in \mathbb{R^+}=[0,\infty) ~and ~n\in \mathbb{N}.$ Define $$\pi(x)= 1 ~~if~~ x\leq n ~and =0 ~if~ x< (n+1).$$ Then for any $x,y\in\mathbb{R},~~ $ I want to ...
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0answers
13 views

Showing boundedness below by sobolev norm

Suppose I know that $h(f)$ is bounded by $h(f) \geq (1 -2ab)\Vert f' \Vert^2_{L^2} - \frac{4a}{b} \Vert f \Vert^2_{L^2}$ with $a,b > 0$ and $f \in H^1$. Then for any $b \leq \frac{1}{2a}$ we can ...
2
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0answers
65 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
2
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0answers
85 views

Prove or disprove that integral term is log-concave

Consider the function $h(x_i, x_j) := \int_{x_i}^{\overline{z}}f(z)F(z-x_i+x_j)dz$, where $f(z)$ is a twice continuously differentiable and strictly positive probability density function defined on $...
1
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2answers
62 views

Prove that $\inf\limits_{f\in X}\,I_a(f)=1-a$.

Let $X:=\big\{f\in \mathcal{C}^1[0,1]\,\big|\,f(0)=0\text{ and } f(1)=1\big\}$, $0<a<1$, and $I_a(f):=\displaystyle\int _0 ^1 x^a \left(\frac{\text{d}}{\text{d}x}\,f(x)\right)^2 dx$. Then prove ...
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0answers
31 views

Semi-bounded operator implies semi-bounded sesquilinear form

So my question is quite simple: If an operator (which corresponds to a sesquilinear form) is semi-bounded (from below) then the sesquilinear form is also semi-bounded (from below)? I'm pretty sure ...
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votes
2answers
260 views

Inequality : $\sqrt {x} - 6 - \sqrt{10} -x \geqslant1$ [closed]

I have solved it by squaring both sides and got inequality $x \geqslant 17/2$ but after that, the solution part have concluded on the equation $4x^2 + 289 - 68x \geqslant4(10 - x)$ How this ...