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Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

18
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0answers
483 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
11
votes
0answers
880 views

How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
6
votes
0answers
333 views

Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
5
votes
0answers
154 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
4
votes
0answers
67 views

Is there analytic solution to this functional problem?

Let $f(x)$ be a function on $[0,+\infty)$. I want to solve the following functional problem: $\min L(f)=\int_0^{\infty} (f'(x))^2\mathrm dx$ subject to $f(0)=a$ $f(x)\ge 0,\forall x$ $\int_0^\...
4
votes
0answers
63 views

Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
4
votes
0answers
92 views

Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and there are $x_1, x_2, \ldots, x_n \in R^d$ such that $$\sum_{i=1}^n (x_i - x') = x - x'.$$ ...
3
votes
0answers
78 views

Can we find an estimate like this: $\left|\int_0^tf\,dg\right|\le C\|f\|_{\infty}\|g\|_{\gamma}$?

Let $f\in\mathcal C^{\lambda}([0,T])$ and $g\in\mathcal C^{\gamma}([0,T])$ (i.e. they are Holder continous), where $\lambda+\gamma>1$. Consider now the following estimate: $$ \left|\int_0^tf\,dg\...
3
votes
0answers
135 views

Question about transportation-entropy inequality (From Villani's book: Optimal Transport, Old and New)

I was reading Villani's book: Optimal Transportation, Old and New. From page 80-83, he introduced some results about dual formulation of transport inequality. Assume $C(\mu,\nu)$ is the optimal ...
3
votes
0answers
151 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(...
3
votes
0answers
65 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
2
votes
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 $$ \|\...
2
votes
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 ...
2
votes
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
votes
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 $...
2
votes
0answers
47 views

inequality involving hypergeometric function

I wish to prove the following inequality: $$A \leq B$$ where $$A=\Gamma(1+\frac{2}{\beta})\cdot \frac{q^{\frac{1}{\beta}}}{1+\beta}\cdot {}_2F_1(1,1+\frac{2}{\beta};2+\frac{1}{\beta};1-q) $$ $$B=(\...
2
votes
0answers
99 views

A conjecture similar to the Hardy inequality

Suppose $\{a_n\}_{n\geq 1}$ is a sequence of nonnegative real numbers. Define sequence \begin{align} b_n = \frac{\sum_{i = 1}^n a_i}{n}. \end{align} Prove the following conjecture: There exists a ...
2
votes
0answers
69 views

Clueless as to how to solve this gamma function differential inequality

Use the conventions $\Pi(x)=\Gamma(x+1)$ and $\frac{d^n}{dx^n}\Pi(x)=\Pi^{(n)}(x)$ henceforth. I am trying to evaluate for what real $\color{blue}{x}>2$ it holds that $$\frac{\Pi^{(\color{red}{n}...
2
votes
0answers
83 views

Solving the inequalities with constraints on function parameters.

Is there any way to show that the following inequality holds for the given functions with constraints? Let $f_1(x) = \frac{1}{2}(\sqrt{(1+(1+a) x+(1-a) y)^2-4 a x (1+x)}+(1-a)x-(1-a)y-1)$, $f_2(x) = ...
2
votes
0answers
63 views

When does $g(x) > 0$ and $0 \le f'(x) \le g(x)f(x)$ imply that $f(x) = 0$?

When does $g(x) > 0$ and $0 \le f'(x) \le g(x)f(x)$ imply that $f(x) = 0$? This question is inspired by my urge to generalize this question: Derivative bounded by the original function I have ...
2
votes
0answers
103 views

Kind of Gronwall Inequality

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$ f(t) \leq A+\int_0^{2t} g(s)f(s) ds $$. Where $f$ and $g$ are as smooth as ...
2
votes
0answers
32 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
2
votes
0answers
88 views

Near-Application of Cauchy-Schwarz Inequality

I have the following situation: I have two estimators of $\alpha$, both via maximum likelihood of the density: $$ f(x,y\mid \alpha,\beta) = f(y \mid x,\alpha,\beta)f(x \mid \alpha) $$ One uses only ...
2
votes
0answers
52 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := \...
2
votes
0answers
46 views

I want to solve the inequality $z′(t)+1≥0$ for all $t≥0$

Let $z(t)$ be a differentiable function for all $t≥0$. I want to solve the inequality: $$z′(t)+1≥0$$ for all $t≥0$. where $z′(t)$ is the derivative of $z(t)$.
2
votes
0answers
861 views

Increasing rearrangement and Hardy-Littlewood inequality

Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it. As you can see in Leoni (or in Lieb & Loss), the decreasing and the ...
1
vote
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 ...
1
vote
0answers
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-...
1
vote
0answers
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 ...
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_{...
1
vote
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 $...
1
vote
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 ...
1
vote
0answers
21 views

Tricks for finding good “close enough” solutions to multivariate recursive relations

As part of an undergraduate project I am attempting to find a solution to this set of recursive inequalities: $$g(n,l,k) \leq g\left(\frac{n-1}{2},l,k-1\right)$$ $$k \cdot g(n,k,l)+\log_2 (n-2^l +1) ...
1
vote
0answers
67 views

$L^2$ convergence and pointwise-norm

Let $\Omega$ be a region, and $f\in L^2(\Omega)$, for simplicity we suppose that $\Omega$ is compact and $f$ is smooth. If we have the inequaliy $$\|E_n\|_{L^2}\leq \frac{C}{\sqrt n}\|f\|_{L^2},$$ ...
1
vote
0answers
29 views

Inequality involving Gaussian kernel

Suppose we have the following implicit functional inequality, $$ |h(x,t)| \leq \int_0^t \int_{\mathbb{R}}\Phi(x-y,s-t)|h(y,s)| \, dy \, ds $$ where $\Phi$ is the Gaussian kernel, $\Phi(x,t) = \...
1
vote
0answers
34 views

Evaluating constants of an inverse estimate

In some analysis on a domain $\Gamma$, I want to employ a type of inverse estimate $$\|F\|_X \le \frac{k}{\Delta{x}}\|F\|_{L^2(\Gamma)}$$ where $F$ belongs to a finite-dimensional subspace, $\|X\|$ is ...
1
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0answers
44 views

Jensens function on the real line exactly and conditional equations f(xy) + f(xy−1) = 2f(x) and f(xy) + f(y−1x) = 2f(x)

What is meant by Jensens function on the real line exactly F(x+y)+F(x-y)=2F(x); jensens equation on a R rather than just a continuousinterval(a closed and bounded real valued (continuum interval such ...
1
vote
0answers
193 views

Strictly Monotonic, Continuous Sub-linear functions with F(1)=1; Any real difference between these and linear functions?

Is there a great deal of difference between sub (or super-linear functions) Functions F, and linear functions, ie, when a sub-linear function $F$, is also strictly monotonic increasing and continuous ...
1
vote
0answers
181 views

Inequalities and limits?

Given a function $f$ such that $\lim \limits_{x \to \infty} f(x) =0$, and want to see if $f(x) >0 $ or $f(x)<0$, but its so hard to tell(its very complicated function). My approach to solve ...
1
vote
0answers
65 views

Best minimum constant for a functional inequality

Let $f$ be a twice differentiable function from $[0,1]$ to $\mathbb{R}$ with $f"$ continuous on $[0,1]$ and $\int_a^b f(x)dx=0$ where $0<a<b<1$. What is the minimum constant $C$, function ...
1
vote
0answers
44 views

A question on $\arctan x$

Let's suppose that I need to know if the angle $2^i \arctan x $ are in the first, second, third or fourth quadrant, for $ i = 1, 2, 3... n$ and some real number $x$. Is it possible to know in which ...
1
vote
0answers
172 views

Defining a convex function geometrically - proof explanation

I am currently reading a book about inequalities, and am on the section about convex functions. It is defined as follows: A function $f : [a,b] \rightarrow \Bbb R$ is called convex in the interval $I=...
1
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0answers
52 views

Lower bound on the difference between max. and min. values of a polynomial over $[-1, 1]$

Problem: $P(x)$ be a monic, n-degree polynomial with real coefficients. Prove that it is not possible that for all $t \in [-1, 1]$, $$\frac{-1}{2^n} < P(x) < \frac{1}{2^n}$$. I tried it to put ...
1
vote
0answers
35 views

A functional equation with an inequality

I have an increasing function on $[0,1]$, $p \mapsto \Pi(p)$, that has the following properties. $$\Pi(0) = 1 - \Pi(1) = 0$$ $$\Pi(p) + \Pi(1-p) < 1 \quad \forall{p} \in (0,1)$$ $$\Pi(p) > p \...
1
vote
0answers
42 views

On the roots of certain functional inequality

I have a question to share, if someone can help me. Let, for each prime $n>2$ $L_{n}:[0,+\infty)\longrightarrow \mathbb{R}$ the function given by $ L_{n}(x):=\sum_{k=1}^{n-1}(-1)^{\Omega(k)}k^{x},$...
1
vote
0answers
43 views

Study the variation of $\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)}$ with $q$

We define function $g(q)$ as the following: $$\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)},$$ where $n^\prime \ge n+1$, $n^\prime \le m$. We note that $q$, $m$, $n$ $n^\prime$ are all ...
1
vote
0answers
55 views

Brunn-Minkowski Inequality : A Partucular Example of a 2 dimensional set.

I have an inequality - $(R_{C+D})^{2} \geqslant (R_{C})^{2} + (R_{D})^{2}$. Representing it as $((R_{C+D})^{4})^{1/2} \geqslant ((R_{C})^{4})^{1/2} + ((R_{D})^{4})^{1/2}$, it follows from Brunn-...
1
vote
0answers
33 views

help me to prove this

I would like to prove that $n/k!$ for $k$ that holds $k\leq \frac{\log n}{\log\log n}$ will always be bigger/equal to $1/2$. I tried to use stirling but got stuck. Any ideas? Thanks, Jonatan
1
vote
0answers
25 views

Non-constant solutions of the functional inequality

Does the functional inequality $$ F(|x|)+F(|y|) \geq \frac{1}{F(|x+y|)} $$ have non-constant solutions $F$? That is, $F(x)\neq \mathrm{const}$. $F$ is a real-valued function of a real variable.
1
vote
0answers
99 views

Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...