Questions tagged [functional-inequalities]

Question 1

Need to find out all the functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $$f(x+y) \leq f(xy).$$ Since $f(x) \leq f(0)$ for every $x \in \mathbb{R}$ and $f(0) \leq f(-x^2)$, it follows that $f(...

Question 2

Suppose I have a sufficiently smooth nonnegative function $y = y(t)$ which satisfies: $$y(t) \le \mathcal{F}(y)(t)$$ for some functional $\mathcal{F}$. Suppose further that $\mathcal{F}(\cdot)(t)$ is ...

Question 3

In book A. Kufner, L. Maligranda and L-E. Persson in part of sufficiency of proof Hardy weighted inequality as $$ ||Hf||_{L_{q,u(x)}} \lesssim ||f||_{L_{p,v(x)}} $$ for $1 \leq p \leq q < \infty $ ...

Question 4

Lately, I was experimenting with composite functions on Desmos when I happened to come across the following observation/claim. Let $f : \mathbb R \to \mathbb R_+$ be defined as $f(x) = x^2 + \frac{1}{...

Question 5

Let $\mathscr{F}$ be a class of functions (each of the form $f: \mathcal{X} \rightarrow \mathbb{R}$ ), and let $\left(X_1, \ldots, X_n\right)$ be drawn from a product distribution $\mathbb{P}=\...

Question 6

Let $0 < s < \eta < 1$ and $p \geq 1$. I would like to prove that there exists a constant $C = C(d, p) > 0$ such that for all $u \in L^p(\mathbb{R}^d)$, the following inequality holds: $$ \...

Question 7

I am studying self-studying PDEs and trying to understand a set of related papers where the inequality the following inequality appears: $N=2$ then $\|u\|_{L^4}^2\leq C \|u\|_{L^2}\|Du\|_{L^2}$. Here $...

Question 8

Let $f:\mathbf{C}\to\mathbf{C}$ be a compactly supported smooth function. Then the following holds: $\lVert \partial_x f\rVert_{L^2}^2+\lVert \partial_y f\rVert_{L^2}^2=\lVert \overline\partial f\...

Question 9

Let $f:[0,1]\to[0,1]$ be differentiable and equal to its own inverse, i.e. $f(f(x))=x$. Assume also that $f(1) = 0$. Does the following inequality hold: $$\left|2\int_0^1\frac{x}{f'(x)}\,dx\right|^3\...

Question 10

This problem is taken off the 2008 problem set for Math 185 in the University of Chicago (PDF link via uchicago.edu) (problem 4 on page 3): Suppose $f$ is analytic on a domain $\Omega \subset \mathbb{...

Question 11

Setting Consider the following setting: Let $d \in \mathbb{N}$. Lorentz space: $L^{p,q} = L^{p,q}(\mathbb{R}^d; \mathbb{R})$, $1<p<\infty$, $1\leq q \leq \infty$ (I assume familiarity with ...

Question 12

Given the following Poincaré inequality: if $f \in C^1([a, b])$ and $f(a) = 0$, then: $$ \int_a^b f^2(x) \, \mathrm{d}x \leq \frac{4(b - a)^2}{\pi^2} \int_a^b f'(x)^2 \, \mathrm{d}x. $$ Therefore, my ...

Question 13

Find all: $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that: $$f(x)f(y)\le f(xy)+x+y, \forall x,y\in \mathbb{R}^+$$ Here is what I have done: Let $x,y \to 1,1$ then: $$f(1)^2\le f(1)+2 \implies f(1)\le 2$$ ...

Question 14

Naive question : Can i get insights of the asymptotic of the derivative of an increasing differentiable function knowing the asymptotic of the function ? For instance if I have $f(x) = o(x^2)$, can I ...

Question 15

Say that I have a continuous increasing function $f : [1, +\infty) \to \mathbb R^+$ which is asymptotically $f(x) = o(x^{1+\varepsilon})$ for any $\varepsilon > 0$ (confer the little-$o$ notation ...

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Questions tagged [functional-inequalities]

Yet another nice functional inequality [duplicate]

Picard iteration and minimal solution

Hardy weighted inequality

Prove that $f^{10}(x) > x^4$, for all real $x$, where $f(x) = x^2 + \frac{1}{5}$.

A trick of symmetrization in proving concentration inequalities

Estimate of the tail fractional seminorms with different exponents near zero

If $N=2$ then $\|u\|_{L^4}^2\leq C \|u\|_{L^2}\|Du\|_{L^2}$

Reference request for an inequality between $L^2$ norms of derivatives

Functional inequality involving function that is its own inverse

What does inequality of complex numbers mean?

Does the fractional Leibniz rule hold in Lorentz spaces?

Poincaré inequality with modified condition [closed]

Find all $f$ such that $f(x)f(y) \le f(xy)+x + y$

Bounding derivative using little o [duplicate]

Yet another functional inequality with asymptotics

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