Questions tagged [integral-inequality]

Question 1

A real-valued function $f \in C[1;2]$ satisfies $\bigg | \int_{1}^{2} f(x) x^n dx \bigg |<2^{-1000n}$ for every positive integer $n$. How to prove that $f(x)=0$ for all $x \in [1;2]$? My attempt: ...

Question 2

If $\alpha$ is a real number in $(0,1)$ and $a_1,a_2, \dots ,a_N \geq 0$, prove that $$\sum\limits_{n=1}^{N} \left(\sum\limits_{k=n}^{N}\frac{a_k}{k} \right)^\alpha \geq \alpha^\alpha \sum\limits_{n=1}...

Question 3

I've been working with the following exercise: Let $f(x)\in C(\mathbb{R})$ and for all $a,b\in\mathbb{R}$, it holds $$ f(a)+f(b)\ge\int_a^b f^2(x)\mathrm{d}x. $$ Show $f\equiv 0$. My attempts so far:...

Question 4

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 5

We have the following inequality for the logarithm: given any $0<a<1$, there exists a constant $C_a$ such that $$\log (1+x) \leq C_a \frac{x}{(1+x)^a} $$ holds for all $x>0$. In other words, ...

Question 6

Let $\ell\ge 2$ be a fixed integer. For each $k\ge1$ define the affine map $$ \theta_{k}(s)=\Bigl(\frac{k}{\ell}+1\Bigr)s-\frac{1}{\ell} =\frac{k+\ell}{\ell}s-\frac{1}{\ell}. $$ Question. Does there ...

Question 7

Precisely, I want to give more details on the following inequalities in here ( page 19) : \begin{align*} \|\partial_y u\|_{L_{x,y}^{\infty}}&\lesssim\sum_{\alpha\in \mathbb{Z}}\|\widehat{\...

Question 8

Let $p\geq 1$ and $u\in L^p(\Bbb R)$, for $t>0$ define $$U(t)= \frac12\int_{\Bbb R} (|u(x+t)-u(x)|^p+|u(x-t)-u(x)|^p) d x.$$ $$U_+(t)= \int_{\Bbb R} |u(x+t)-u(x)|^p d x.$$ Then we find that $(0,\...

Question 9

Young's Convolution Inequality states that, on any group $G$ with some defined $L^p$-norms, if $f\in L^p(G)$ and $g\in L^q(G)$ and $\frac1p + \frac1q = \frac1r + 1$, then $$ \|f\ast g\|_r \le \|f\|_p \...

Question 10

I am dealing with a PDE on $H^1(\mathbb{R})$ and it turns out that considering functions functions $f\in H^1(\mathbb{R})$ such that $$\|f\|_2\leq \|f'\|_2\;\;\;(1)$$ might be useful. While I ...

Question 11

Let $f$ be a twice differentiable function on $[0,2]$ such that: $$ f(0)-(a+1)f(1)+af(2)=1, \quad f'(2)=0$$ For some $a >0$. Prove that: $$\int_0^2 [f''(x)]^2 dx \geq \frac{3}{a^2-3a+4}.$$ My ...

Question 12

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 13

Let $f \geq 0$ and $f \in L^1([0,1])$. Assuming that all integrals exist, I want to show that $$\int_{[0,1]} f dx \int_{[0,1]} \log(f) dx \leq \int_{[0,1]} f\log(f) dx .$$ One thing which I though of ...

Question 14

Prove that there exists a positive constant $C$ such that $$\int \limits_{0}^{\infty} e^{-y}[u(y) - \langle u \rangle]^2 \mathrm{d}y \le C \int \limits_{0}^{\infty} e^{-y}[u'(y)]^2 \mathrm{d}y$$ for ...

Question 15

I am trying to show that if $f \in L^2([0,1])$ with $||f||_{L^2}\leq 1$, then $$\int_0^1\frac{x}{(1+|f(x)|)^2}dx\geq \frac{1}{9}.$$ My ideas so far: $\int_0^1\frac{x}{(1+|f(x)|)^2}dx\geq \int_0^1\...

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Questions tagged [integral-inequality]

How to prove that $f \equiv 0$ on $[1;2]$?

Maybe an integral inequality in discrete form?

Show a continuous function over $\mathbb{R}$ is zero [duplicate]

Hardy weighted inequality

interpolation type inequalities for log or certain convex functions

Uniform bound for a family of integrals with affine rescaling inside the integrand

How to bound a periodic function's first derivative by its $H^3$ norm and the $L^2$ norm of $\Delta u$?

On Chebychev type inequality for Lp-first oder difference

Optimal constants in Young's Convolution Inequality for spaces other than $\mathbb{R}^n$

Subset of $H^1(\mathbb{R})$

$\int_0^2 [f''(x)]^2 dx \geq \frac{3}{a^2-3a+4}$

Estimate of the tail fractional seminorms with different exponents near zero

Integral inequality with logarithm

Proving a weighted Poincaré inequality

Estimating integral from below

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