Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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1answer
139 views

A question on (odd) perfect numbers

(Note: This has been cross-posted to MO.) Let $\sigma(x)$ be the (classical) sum of the divisors of $x$. A number $N \in \mathbb{N}$ is called perfect if $\sigma(N)=2N$. An even perfect number $U$ ...
2
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2answers
52 views

$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy \leq ||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $

We are given the following, $$ b:\mathbb R^d \times \mathbb R^d \rightarrow \mathbb R,\;\; f:\mathbb R^d\rightarrow \mathbb R $$ and $$ f\in L^2(\mathbb R^d)\; ,\;b\in L^2(\mathbb R^d\times \mathbb R^...
2
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1answer
88 views

Proving inequality $(x^2+y^2)(y-1)+yx-y^2<0$

I have an inequality which came out of Lyapunov function for system of ODE's: $$(x^2+y^2)(y-1)+yx-y^2<0.$$ To prove stability of my solution, I have to prove that the inequalty is true in area $0&...
2
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1answer
91 views

Inequality with Bessel Functions of the first kind

How can be proven the following inequality: $$\int_0^1dx|J_k(x)'J_k(x)|\lt\frac{1}{2}\int_0^1dx|J_k(x)'^2|$$ where obviously: $J_k(x)'=\frac{d}{dx}J(k,x)$? Thanks.
2
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1answer
101 views

Are there (known) bounds to the following arithmetic / number-theoretic expression?

I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know): Are there (known) ...
2
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1answer
194 views

Inequality for Gamma functions

Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true? $$ \frac{2^{m-k}\Gamma(n+1)\Gamma\left(\left[\frac{m+1-k}{2}\right]\right)\Gamma(m+1-n)}{\Gamma(m+1)\...
2
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0answers
69 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(...
2
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0answers
24 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\...
2
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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 $N((2,0),2)$ ...
2
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0answers
51 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
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0answers
86 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
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0answers
71 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
88 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
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1answer
76 views

Optimal constant for a functional inequality

I've been working on a problem for a while now and can't seem to arrive at a solution. I have to find the optimal constant $C$ that satisfies the inequality: $$Cu(0)^2 \le \int_0^1u(x)^2\ dx+\int_0^...
2
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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=(\...
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0answers
101 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
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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}...
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0answers
84 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
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0answers
64 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
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0answers
37 views

Proof for convexity of a function [duplicate]

A continuous function is called convex on an interval $I\subseteq\mathbb R$ iff $$\forall x,y\in I\;\forall t\in[0,1]:f\big((1-t)x+ty\big)\leq (1-t)f(x)+tf(y).$$ I want to prove that this is ...
2
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0answers
92 views

Minimum conditions for $f(x) + f(y) < f(x+y)$ [closed]

I would like to know what the minimum conditions on a function $f$ are so that $$f(x) + f(y) < f(x+y)$$ for any $x,y > 1$? Also a proof(/proof idea/reference for a proof) would be great. Thanks :...
2
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0answers
106 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
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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
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0answers
92 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
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1answer
118 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
2
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0answers
53 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
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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
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2answers
765 views

Convexity of log sum function

Is $f\left( x \right)=\log \left( \sum_i \beta_i e^{-\alpha_ix} \right)$ a convex function where $\beta_i,\alpha_i\in \mathbb{R}$?
2
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0answers
885 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 ...
2
votes
1answer
99 views

Inequality, with quotient substitution

I do not know how to prove this inequality: Suppose that $x_i>0$ and $x_1\cdot ...\cdot x_n=1$, show that $$\frac{1}{1+x_1+x_1x_2}+...+\frac{1}{1+x_n+x_nx_1}>1$$ The hint is to use quotient ...
2
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0answers
331 views

An application of Poincare inequality [solved] [duplicate]

I am woking on Evans PDE problem 5.10. #15: Fix $\alpha>0$ and let $U=B^0(0,1)\subset \mathbb{R}^n$. Show there exists a constant $C$ depending only on $n$ and $\alpha$ such that $$ \int_U u^2 dx \...
2
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3answers
52 views

RATIONAL INEQUALITY - Find the values of a such that range of $f(x)=\frac {x+1}{a+x^2}$ contains $[0,1]$

Find the values of $\text{“}a\text{''}$ such that range of $f(x)=\frac{x+1}{a+x^2}$ contains $[0,1]$ where am i wrong ??: I took $2$ cases : Case $1$ : $a+x^2 \gt 0$ , I solved the inequality $f(x)...
1
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4answers
226 views

How to prove (approximate) inequalities involving exponential (logarithms) and polynomials.

How to prove the following inequality $$(1-x) \ln(1-x) \geq -x + x^2/2\, \textrm{ for } x\in(0,1)?$$ In general how should one go about proving/approximating exponential and logarithms to get the ...
1
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2answers
8k views

Quadratic Absolute Value Inequality

Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
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 ...
1
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2answers
61 views

How to minimise this expression?

I came across this question in a reference book: $x,y,z$ are positive reals such that $x^3\cdot y^2\cdot z^4=17$. Find the minimum value of $3x+5y+2z$. Well, I know that the expression containing ...
1
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2answers
61 views

Find all $x$ to satisfy $|x-a|<|x-b|$.

Exercise: Using signs of inequality alone (not using signs of absolute value) specify the values of $x$ which satisfy the following relation. Discuss all cases. $$|x-a|<|x-b|$$ Solution: $|x-...
1
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3answers
98 views

What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$? A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here. Lastly, note that ...
1
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1answer
52 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{...
1
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1answer
89 views

Proving $\int_0^1 |f'(x)-f(x)|dx \ge 1/e $

Let f be a differentiable function on (0,1) inclusive, such that f(0)=0 and f(1)=1. If the derivative f' of f is also continuous on (0,1) inclusive, prove that $$\int_0^1 |f'(x)-f(x)|dx \ge 1/e $$ ...
1
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1answer
2k views

Prove that $\sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + \frac{1}{x}}$ for $0 < x < 1.$

Prove that $$\sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + \frac{1}{x}}$$for $0 < x < 1.$ I also received this hint: With the square root in the left-hand side, you may be tempted to ...
1
vote
2answers
914 views

Find the smallest integer $n$ such that $(x^2 + y^2 + z^2)^2 \leq n(x^4 + y^4 + z^4)$ for all real numbers $x, y,$ and $z.$

Find the smallest integer $n$ such that $$(x^2 + y^2 + z^2)^2 \leq n(x^4 + y^4 + z^4)$$for all real numbers $x, y,$ and $z.$ How should I manipulate this inequality? I am stuck and don't know how ...
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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
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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2answers
52 views

How to Prove that this Function is Constant?

Let be $h: \mathbb{R} \rightarrow \mathbb{R}$ and c a positive constant, if $\forall a, b \in \mathbb{R}$ we have that: $$\frac{|h(a)-h(b)|}{a-b}\leq |a-b|^c$$ Prove that $h$ is constant. I ...
1
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1answer
72 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
1
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2answers
70 views

Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
1
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1answer
64 views

$ {\|f\|}_p = \sqrt[p]{\int_{a}^{b} |f(x)|^p {\rm d}x}$ is a norm

Consider the space $C([a,b])$ of all continuous functions $f\colon [a,b]\rightarrow \mathbb{R}.$ Show that the function $\|\cdot\|_p\colon C([a,b]) \rightarrow [0,\infty),p>1$, given by $$ {\|f\|}...
1
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1answer
88 views

Does this function have a (global) minimum?

A good day to everyone. Does the following function have a (global) minimum: $$1 + \frac{1}{x} + {\left(1 + \frac{1}{x}\right)}^\theta,~~x\in\mathbb R$$ where $$\theta = {\displaystyle\frac{3\log ...
1
vote
1answer
33 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, ...