Questions tagged [integral-inequality]
For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
1,123
questions
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An integral inequality involving the Bernoulli polynomials
The classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{z\operatorname{e}^{t z}}{\operatorname{e}^z-1}=\sum_{j=0}^{\infty}B_j(t)\frac{z^j}{j!}, \quad |z|<2\pi.
\end{...
- 1,846
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votes
2
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Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$
Let $\Omega=[0,1]^d$ for some $d\ge 1$, and let $w:\Omega \to (0,\infty)$ be a continuous function.
Is is true that $$\Vert w f \Vert_{L^2(\Omega)}^2+ \left\Vert \frac{1}{w}g \right\Vert_{L^2(\Omega)}^...
- 4,928
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Local property of an integration inequality to global result
Here is the question.
Let $f$, $g$ be locally integrable functions on $\mathbb{R}^n$ such that
$$\inf_{a \in \mathbb{R}} \int_{B} |f(x) - a|\ dx \leqslant \int_B|g(x)|\ dx$$
for all balls $B$ in $\...
- 1
1
vote
1
answer
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Determine the convexity of a ball in a metric space.
Let $V$ be the set of all Lebesgue integrable functions. $V$ forms a vector space with respect to general function addition and scalar multiplication. Let $X \subset V$ is the set of positive and ...
- 102
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answers
61
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The constant in Gronwall's inequality
The classical Gronwall's inequality is as follows:
Assume that
$$
f(t)\leq K+\int_0^tf(s)g(s)ds,
$$
where $f(t)$ and $g(t)$ are continuous functions in $[0,T],$ $g(t)\geq 0$ for $t\in [0,T],$ and $K\...
- 61
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Integral inequality and Riesz Kernel
I am currently working on some Fractal Geometry, specifically how the Riesz capacity and the Fourier transform can inform us about the Hausdorff dimension. My book states the following inequality as ...
- 13
1
vote
1
answer
69
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Sign of product of two integrals
Let $\Omega$ be an open bounded regular subset of $\mathbb{R}^N$. Let $\lambda_k$ with $0<\lambda_1<\lambda_2\leq\lambda_3\leq...\uparrow\infty$ be the sequence of eigenvalues of $-\Delta$ in $\...
- 123
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18
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Question about the proof that uniform asymptotic stability can be characterized by KL function. (Lemma 4.5 in Nonlinear Systems (3rd) by Khalil)
Lemma 4.5 in Nonlinear Systems (3rd):
Consider the nonautonomous system
\begin{equation}
\dot{x} = f(t,x) ,
\end{equation} where $f : [0,\infty) \times D \to \mathbb{R}^n$ is piecewise ...
- 11
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Finding a condition to bound $f$ satisfying an integral inequality
I was concerning the following problem:
Let $f$ be a continuous function on $[0,\infty)$ such that $f(x)\ge 0$, $f(0)\ne 0$. Let $g$ be an integrable function on $[0,\infty)$ such that $$\int_0^\...
- 569
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Inequality with distribution and integral
Let $a \in (0,1)$ be the (unique) solution of:
$\displaystyle \int_0^1 (\theta - a)e^{\dfrac{(\theta - a)^2}{\beta}}g(\theta)d\theta = 0$ (1), where $g(\theta)$ is a continuously differentiable ...
- 31
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vote
0
answers
81
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Reversed form of Grönwall's inequality?
I am looking for a "reversed" form of Grönwall's inequality. Let's recall the usual requirements from Grönwall's inequality. First, denote by $I\subset\mathbb{R}$ an interval of the form $[a,...
- 1,147
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Relation between two norms
Let $p\ge 2$ and $w:\mathbb{R}^d\to \mathbb{R}_+$ be a weight function normalized such that $\|w\|_{L^1}=1$ (the examples I have in mind would be a Gaussian or a two-sided exponential for example).
...
- 61
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vote
3
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How to prove this inequality involving trigamma functions?
While solving a problem I succeeded to reduce it to the following inequality:
$$
\forall \{a,b,z\in\mathbb R_+,\ a\ne b\}:\quad 0<\frac1{a-b}\int_0^\infty\frac{t(a^2e^{-azt}-b^2e^{-bzt})}{1-e^{-t}}...
- 26.7k
1
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1
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Integral inequality with exponents
Let $f(x):[0;1]\to\mathbb{R}$ be continuous function. Prove that $$\int_0^1e^{f(x)}dx \cdot \int_0^1e^{-f(x)}dx \geq 1+\int_0^1(f(x))^2dx-\Bigg(\int_0^1f(x)dx\Bigg)^2.$$
I tried to use Cauchy–Schwarz ...
- 169
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Can we prove Young's Convolution Inequality only using Interpolation?
I tried to prove Young's Convolution Inequality $\|f\ast g\|_r\leq \|f\|_p \|g\|_q$ for $1/r+1=1/p+1/q$, where $p,q,r\in[1,\infty]$. From the Riesz–Thorin Interpolation theorem, it suffices to prove ...
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