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Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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Condition for $s(t) + 2 \int_0^\infty s(\tau)s'(\tau+t) d\tau \geq 0$

Let $s$ be a function defined on $\mathbb{R}^+$ such that $s(t)>0$ $s(0) = 1$ $s'(t)<0 $ I am trying to find minimal conditions on $s$ for the inequality $s(t) + 2 \int_0^\infty s(\tau)s'(\tau+...
hugues_myr's user avatar
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A Hölder bound on an integral involving the complex exponential.

I want to show that if $\gamma \in (0,1)$ we have: $$\int_{\mathbb R^n}\frac{|e^{i x \cdot \xi}-e^{iy\cdot \xi}|^2}{|\xi|^{n+2\gamma}}\,\mathrm d \xi \le C_{n , \gamma} |x-y|^{2 \gamma}$$ for some $C_{...
slowlight's user avatar
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Gradient of a radial function

In the lecture on Gagliardo-Niremberg inquality, there was mentioned the fact that for functions: $$f_{\lambda}(x) = (\lambda + \vert x \vert^q)^\frac{p-n}{p}, \; \text{ where } \frac{1}{q}+\frac{1}{p}...
Pingu's user avatar
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3 votes
1 answer
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Non-decreasing functions satisfying functional inequalities $f(1+ax)\leq a f(1+x)$ and $f(xy)\leq f(x)+f(y)$

Can we exactly determine a class of non-decreasing functions defined on the set of non-negative real numbers satisfying: $$f(1+ax)\leq a f(1+x)$$ and $$f(xy)\leq f(x)+f(y)$$ for any $x,y\in [0,\infty)$...
Marija's user avatar
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2 votes
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78 views

Dense subset of $GL(n)$ and existence of elements with some properties.

In the following $|.|$ denotes the operator norm in the space $GL(n):=$set of invertible matrix of size $n \times n$. I have the next problem: Let $S \in GL(n)$ and $\mathcal{G}$ be a dense subset of $...
C L 's user avatar
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1 vote
1 answer
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(Dis)prove a statement similar to Bessaga fixed point theorem

Prove or disprove that if $f:\Bbb R\to\Bbb R$ is such that $|f(x)-f(y)|<|x-y|$ for all different real numbers $x$, $y$, then $f(x)=x$ has a unique solution. Firstly, there can't be $f(a)=a$ and $f(...
youthdoo's user avatar
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3 votes
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77 views

What functions satisfy $f(ax) - f(a(x-1)) > f(b(x+1)) - f(bx)$ for all $a, b \in \mathbb{R}^+$ and $x \in \mathbb{Z}^+$.?

I am looking at a family of functions $f : [0, \infty) \rightarrow [-\infty, \infty)$ satisfying the following property: $$f(bx) - f(b(x-1)) > f(a(x+1)) - f(ax) \quad \text{for all $a, b \in \...
K.C.'s user avatar
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A challenging problem in ODEs

Problem. Let $f:D\to\mathbb R,$ be continuous, where $D\subset\mathbb R^2$, open, and let $\varphi,\psi :[\tau:T]\to\mathbb R,$ differentiable functions. If $\varphi(\tau)=\psi(\tau),$ and $$ \varphi'(...
Yiorgos S. Smyrlis's user avatar
2 votes
0 answers
126 views

Functional Calculus Inequality?

When I was solving an olympiad problem, I came out of this hypothetical inequality: For bijective $f:[0,1]\rightarrow [0,1]$, continuous in $[0,1], f(0)=1, f(1)=0$, then we have: $$\int_0^1 xf(x)dx\...
MafPrivate's user avatar
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2 answers
80 views

Some inequality I stumbled upon, $b^3-a^2b+2a \geq 0$

$b^3-a^2b+2a \geq 0$ Given: $(a,b) \in \mathbb{N}$ and a>b. Find when this holds. This is an inequality I stumbled upon while I was surfing, and like any other day, I sat down trying to think about ...
The Revolution's user avatar
1 vote
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45 views

Prove that one function is larger than another

I'm trying to prove that $u_1(x) \geq u_2(x)$ for $ x \in [0,1] $. $$u_1(x) = \frac{1}{2} \cdot h^{-1}\left(\frac{1}{2} \cdot h(v(1)) + \frac{1}{2} \cdot h(v(x))\right) + \frac{1}{2} \cdot h^{-1}\left(...
Chris's user avatar
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2 answers
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Find functions $f(x)$ and $g(x)$ such that the following conditions are satisfied for all $x > 0$: [closed]

$0 < f(x) < 1$, $g(x) < \frac{f(x)}{x} < c$ for some constant $c$, $\frac{d}{dx}g(x) > 0$.
Pankaj Mishra's user avatar
1 vote
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Is there a convex difference estimate which depends only on non-matching components?

I have a convex function $f$ on $\mathbb{R}^n$ and a collection of $m\in\mathbb{N}$ convex weights $\{w_i\}_{i=1}^m\subset\left]0,1\right]$ satisfying $\sum_{i=1}^mw_i=1$. I have two vectors $(x_1,\...
grouse's user avatar
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2 votes
1 answer
80 views

Is this following inequality with increasing powers of the components true for small $x$? And if yes, what's the positive constant $C?$

Let $0< k_1\le k_2\dots \le k_m, k_i \in \mathbb{N}, k_1 \text{ an even positive integer }. f(x_1\dots x_m):=\sum_{i=1}^{m}{x_i}^{k_i}.$ I wanted to prove, if possible, that in a small enough ball $...
Learning Math's user avatar
3 votes
2 answers
237 views

Prove $f\left(nx\right)\leq nf\left(x\right).$

Probelm, $f:\left [ 0,+\infty \right ) \to \mathbb{R}$ is continuous on $\left [ 0,1 \right ]$ and differentiable on $\left ( 0,1 \right ) $, and $f(x) = f(x+1),\; f(0) = 0$, $f^{\prime}(x)$ is ...
Zhiwei's user avatar
  • 117
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1 answer
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$L^2$ norm of $f(x) - f(y)$

Let $W(x,y) \in L^\infty(\Omega^2)$. Is there a way to get a good lower bound on $\int_{\Omega^2} W(x,y) (f(x) - f(y))^2 dydx$ in terms of $\int_\Omega W(x,y) f(x)^2 dx$? We can say \begin{align} \...
900edges's user avatar
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2 votes
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Find all possible values of $H(x,y,z,t) = \frac{x}{t+x+y}+\frac{y}{x+y+z}+\frac{z}{y+z+t}+\frac{t}{z+t+x}$ if $x, y, z, t > 0$.

If $x, y, z, t > 0$, find all possible values of $H(x,y,z,t) = \frac{x}{t+x+y}+\frac{y}{x+y+z}+\frac{z}{y+z+t}+\frac{t}{z+t+x}$. How I think this can be solved: First off, note that $H$ is an ...
Prominens's user avatar
1 vote
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70 views

From pointwise inequality to the inequality in the sense of distribution

Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous on $\mathbb{R}^n$. If $u\in C^\infty(E^c)$ and there is a positive ...
William's user avatar
  • 11
2 votes
0 answers
47 views

A possible upper bound for a function that satisfies a singular integral inequality

I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality: $$ |v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left( |...
Taki Zeg's user avatar
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147 views

Understanding the union of two sets inequality in Desmos...

I was looking for a way of graphing the union of two sets and found the following in Desmos - Union of Sets Desmos. I've also included an image Image of Desmos page My question is what is where has ...
M W's user avatar
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6 votes
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166 views

A lemma in the application of Concentration compactness principle in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
IMOS's user avatar
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Functional inequality $f(x)≥f(x+yf(x))(y+1)$

Show that if function $f$ satisfies $f(x)\geq f(x+yf(x))(y+1)$ for $x,y>0$ then $f(x)>0$ is false. It is clear that $f$ is non-increasing but I can't show that it will pass OX and not assuming $...
Rhegf2wffw's user avatar
3 votes
1 answer
136 views

Analysis of the functional inequality $f(x-s)f(y+s) \geq f(x)f(y)$

I would like to find references or analysis of the two functional inequalities: $f(x-s)f(y+s)\geq f(x)f(y)$ $f(x-s)+f(y+s) \geq f(x) + f(y)$ where $x,y\in \mathbb{R}$, $y>=x$ and $s>0$. ...
Hushus46's user avatar
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2 votes
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41 views

Semigroup property between pseudodifferential operators and differential operators

Given a positive integer $n$ and a>0. Let consider the operators $\nabla^n (\cdot)= \sum_{i=1}^d \partial_i^{n}(\cdot) $, and $(1- \Delta)^{\frac a 2} $ defined at the Fourier level as (modulus ...
g.cooper's user avatar
2 votes
0 answers
107 views

Functional inequality $f(x)\cdot \cos( x-y)\le f(y)$.

I am doing special manual of functional equations. While i was searching problems for my book, i have found this: Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x) \cdot \cos(...
Dmitry Ch's user avatar
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0 answers
42 views

Difference of two sub additive function

A real valued function $ f : \mathbb{R} \to \mathbb{R} $ is said to be sub additive if $ f( x+y ) \leq f( x ) + f( y ), \quad x,y \in \mathbb{R}. $ There are some nice properties related to sub ...
swapan Jana's user avatar
2 votes
0 answers
51 views

Function that is bounded above by an integral expression is identically zero

Question: Let $\phi(t)$ be a positive continuous function on $[0, \infty)$ and $f(t,x)$ be continuous function of two variables so that $|f(t,x)| \leq \phi(t) |x|$. Suppose $\int_0^\infty \phi(t) < ...
L-JS's user avatar
  • 715
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0 answers
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linear transformation of a subset with inequality

Let $F$ be the vector space of all functions $R \to R$. Let $U_2$ be the subset of F formed by all functions bounded by above (a function $f: R \to R$ is bounded from above if there exists $M ∈ R$ ...
Rocky's user avatar
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1 vote
1 answer
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Inequality in the unit ball of Sobolev Space $W^{1,1}(\mathbb{R})$

In the book Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis The Sobolev space $W^{1,p}(I)$ is defined to be $$W^{1,p}(I)=\{u\in L^{p}(I);\exists g\in L^p(I)\text{ ...
John He's user avatar
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1 answer
142 views

Asymptotics for $f(x)$ such that $f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$?

Consider $$f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$$ $$f(n) < \pi(n+1)$$ Where $\pi$ is the prime counting function and that inequality follows for instance from these here $f(x) + f(x/2) + f(x/3)...
mick's user avatar
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1 vote
0 answers
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Show that $M(f * g) \leq (M f) * (M g)$ for all $f, g \in L^1(R)$

Let $f \in L^1 (R)$. By $M f$ be the restricted maximal function defined by $$(M f )(x) = \sup_{0 <t<1} \frac{1}{2t} \int_{x-t}^{x+t} |f(z)| dz.$$ Show that $M(f * g) \leq (M f) * (M g)$ for all ...
Mr. Proof's user avatar
  • 1,525
0 votes
2 answers
87 views

Inequalities and the growth of hypergeometric functions with respect to the coefficients

For a question in geometry I was led to the following questions on hypergeometric functions. For $a>0$, let $$h_a(x)={}_2F_1(a i, -a i; 1; x)$$ where $i$ is a root of $-1$. Here $h_a(1)=\frac{\sinh ...
Han's user avatar
  • 115
2 votes
0 answers
79 views

Maximal Value of Integral $\int_{-3}^3 \frac{f(x)}{x^3}\;dx$

Given a twice differentiable function $f:(-3,3)\to \mathbb{R}$ such that $$\frac{(x-3)f''(x)-3f'(x)}{x^4}+\frac{12f'(x)}{x^5}\le 2.$$ Find the maximum value of $\int_{-3}^3 \frac{f(x)}{x^3}\;dx$. I ...
Wildan B. W.'s user avatar
1 vote
1 answer
35 views

Last step in $\nabla_z f(x,y,z), \partial_y f(x,y,z)\le c(1+|y|^{p-1}+|z|^{p-1})$ imply $ f(x,y,z)\le c(1+|y|^p+|z|^p)$

Let $\Omega \subset \mathbb{R}^n$ open and bounded with Lipschitz boundary $\partial \Omega$ and the functional $$I(u)=\int_\Omega f(x,u(x),\nabla u(x))dx$$ with $f: \bar\Omega\times\mathbb{R}\times ...
some_math_guy's user avatar
8 votes
1 answer
351 views

a functional inequality (positive functions on circle)

I am trying to prove the following inequality but have no idea at all. Does anyone have any idea how to tackle it or know some related sources? $$\frac{\int_0^{2\pi}(\frac{1}{2}\frac{f^2}{g}\frac{dg}{...
Richard's user avatar
  • 97
1 vote
0 answers
97 views

How to prove the Poincaré inequality by induction on the dimension?

Let $\Omega$ be a bounded connected open subset of $\mathbb{R}^n$ with Lipschitz boundary and $1\leq q \leq \infty$. By the Poincaré inequality, there is $c>0$ such that $$\|u\|_{L^q(\Omega)} \leq ...
Marcos's user avatar
  • 11
4 votes
1 answer
93 views

Integral inequality involving polynomial

I saw the following inequality on another website without any solution: for any $n\ge 0$, show that $$\min_{a_1,a_2,\cdots,a_n\in\mathbb{R}}\int_0^{+\infty}(1+a_1x+a_2x^2+\cdots+a_nx^n)^2e^{-x}dx=\...
Ivan's user avatar
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0 votes
0 answers
74 views

Prove that $(\frac{\sin x}{x})^3\geq{\cos x}$ with $x\in [0,\, \frac{\pi}{2}]$ [duplicate]

Let $x$ be a real number satisfying $0\leq{x}\leq{\frac{\pi}{2}}$. Prove that $(\frac{\sin x}{x})^3\geq{\cos x}$. I have a problem I need help. The idea of ​​the proof of this problem is to use ...
Question 's user avatar
0 votes
0 answers
108 views

A question about the changes of $f$ when an inequality between $f(x,y)$ and $g(x,y)$ is satisfied

I have two independent, continuous functions that satisfy the following inequality $$f(x,y) \ge g(x,y).$$ Now suppose $g(x,y)$ is set to be fixed, then $x$ and $y$ are constrained. Both functions are ...
Loop Corrections's user avatar
0 votes
1 answer
49 views

Appropriate exponent for inequality [closed]

Let consider $1<b<2$, the sets $A=\{ x: |x|\geq2 \} \subset \mathbb{R}^3$, $B=\{x: |x|<2 \} \subset \mathbb{R}^3$ and the expression \begin{equation} |x|^{1-b} \chi_{ A} + |x|^{2-b} \chi_{ B ...
g.cooper's user avatar
2 votes
1 answer
55 views

Bellman-like integral inequality

I have a weakly decreasing, continuous function $w(\cdot)\geq 0$ on $[0,T]$, with $w(0)=1$. I also have a continuous, decreasing function $b(\cdot)$, where $b(t)\geq 0$ for $t\in[0,T]$. I know that \...
Ralph 's user avatar
  • 31
-2 votes
1 answer
133 views

Popoviciu's inequality

If $f(x): \mathbb R \to \mathbb R$ is a convex function, prove that $$f(x) + f(y) + f(z) + 3 f(\frac{x + y + z}{3}) \geq 2 f(\frac{x + y}{2}) + 2 f(\frac{x + z}{2}) + 2 f(\frac{y + z}{2})$$ I proved ...
math.enthusiast9's user avatar
1 vote
3 answers
70 views

Proof of inequality $x + x^{-1} - x^r - x^{-r} \geq 0$ for $x>0$ and $0<r<1$.

I am trying to prove the inequality $$x + x^{-1} - x^r - x^{-r} \geq 0$$ for $x>0$ and $0<r<1$. I believe this to be true as it was cited in a proof that I followed, but was not proved there. ...
Will Ford's user avatar
0 votes
1 answer
27 views

Do injection preserves inequalities?

Suppose $f$ is injective; is it true that $$ \Pr[f(x)<y] \qquad \text{is equivalent to} \qquad \Pr[x<f^{-1}(y)]? $$ provided $y\in\mathrm{Ran} f$? What if $f(x)=-x$?
ric.san's user avatar
  • 121
1 vote
0 answers
43 views

Function satisfying $\alpha x(2-x) + (1-\alpha x)g(x) < g(x-\alpha x)$ for all $0\leq x \leq 1$ and $0\leq \alpha <1$.

I am looking for a function $g(x) \in [1, c], c\geq 1$ that satisfies $$\alpha x(2-x) + (1-\alpha x)g(x) < g(x-\alpha x)$$ for all $0\leq x < 1$ and $0\leq \alpha <1$. Across such $g$, I want ...
AspiringMat's user avatar
  • 2,447
1 vote
1 answer
90 views

Functional Inequality. Is my approach ok?

Let $f: \mathbb R \to \mathbb R$ be a differentiable function such that $$f(x)^2+f'(x)\le 0.$$ Show that the zero function is the only solution. I started bwoc assuming that there is a point $x$ such ...
Eduard Valentin's user avatar
4 votes
1 answer
117 views

An inequality considering linear ODE

Let $p(t),q(t)$, and $r(t)$ be continuous functions on an open interval $I$ and let $t_0\in I$. Assume that there exists a positive constant $M$ such that $$|p(t)|+|q(t)|+|r(t)|<M$$ for all $t\in I$...
mathhello's user avatar
  • 918
4 votes
1 answer
151 views

Increasing function $f(x,y,y)=f(x,x,y)$ implies $f(x,y,z)=g(x,z)$?

Got a super super interesting contest problem. $f(x,y,z)$ is a continuous function of three variables; it is required that $x\leq y\leq z$ (this means that $f$ is only defined on the 3-tuples such ...
High GPA's user avatar
  • 3,776
4 votes
1 answer
211 views

Proving or disproving the functional inequality

I'm trying to prove or disprove the inequality $$\int_{\mathbb{R}}f(x)^{p-2}f'(x)^4dx\leq \frac{3}{p(p-1)}\int_{\mathbb{R}}f(x)^pf''(x)^2dx$$ for any real-valued function $f$ (provided the RHS is ...
bellcircle's user avatar
  • 2,949
1 vote
2 answers
119 views

Solve the functional inequality $f(x+1)>(f(x))^2-f(x)+1$

Suppose $f(x):[0,\infty)\longrightarrow [0,\infty)$ satisfies $$f(x+1)>(f(x))^2-f(x)+1$$ for sufficiently large values of $x$. Can we solve this functional inequality or get some good information ...
stephan's user avatar
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