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Questions tagged [integral-inequality]

For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.

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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{...
qifeng618's user avatar
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2 votes
2 answers
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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)}^...
Tulip's user avatar
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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 $\...
ZYZ's user avatar
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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 ...
Mixi Andrew's user avatar
2 votes
0 answers
61 views

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\...
Rayyyyy's user avatar
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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 ...
Requaero's user avatar
1 vote
1 answer
69 views

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 $\...
Mathslover's user avatar
1 vote
0 answers
18 views

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 ...
Lau's user avatar
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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^\...
MathLearner's user avatar
2 votes
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135 views

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 ...
Dave299's user avatar
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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,...
Satana's user avatar
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1 vote
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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). ...
mrry0's user avatar
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3 answers
81 views

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}}...
user's user avatar
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1 vote
1 answer
104 views

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 ...
perenqi's user avatar
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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 ...
Confusion's user avatar
2 votes
0 answers
116 views

How can we derive this integral inequality?

Furthermore, let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $\mu$ be a probability measure on $(E,\mathcal E)$; $\zeta$ be a Markov kernel on $(E,\mathcal E)$; $\pi$ be a ...
0xbadf00d's user avatar
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4 votes
2 answers
114 views

Integral inequality with specific condition

$f(x)$ is integrable on $[0;1]$ and $\int\limits_0^1 f(x) dx = \int\limits_0^1 xf(x) dx = 1$. Prove that $$\int\limits_0^1(f(x))^2 dx\geq4$$ I tried to use Cauchy–Schwarz integral inequality. I tried ...
perenqi's user avatar
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5 votes
1 answer
171 views

Integral inequality on $[0,1]$

I found the following question on another forum and unfortunately I don't have any additional information about it: Let $f$ be non-negative and square integrable on $[0,1]$. Prove that $$\left(\int_0^...
Ivan's user avatar
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Finding a constant to bound a function

I'm working on the following problem: Let $f$ be a continuous function defined on $[0,\infty)$ such that $f(x)\ge 0$ and $f(0)\ne 0$. Let $g$ be a function defined on $[0,\infty)$ such that $\int_{0}^...
MathLearner's user avatar
5 votes
2 answers
206 views

$u''+u\geq 0$ then $\int_{0}^{2\pi} (u'(x))^2 dx \leq \int_{0}^{2\pi} (u(x))^2 dx.$

Let $u: \mathbb{R} \rightarrow \mathbb{R}$ be a $2\pi$-periodic and smooth function so that $u''(x) + u(x) \geq 0$ for any $x\in\mathbb{R}$. I want to prove that $$\int_{0}^{2\pi} (u'(x))^2 dx \leq \...
Hanh's user avatar
  • 279
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1 answer
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Inequality for minimum of two random variables

I am working with two non-decreasing functions, $T_1(t)$ and $T_2(t)$, where $t$ represents a parameter that is influenced by randomness, $t\in[0,1]$. Specifically, I am trying to prove an inequality ...
Arseniy Gorbushin's user avatar
1 vote
1 answer
134 views

Hard Definite integral inequality involving logarithms and polynomials

Prove that if $f(x) = lnx$ , $\int_{1}^{e^2}\frac{f(xe^{x+1})dx}{(x+1)^2\ +\ f^2(x^x)} > \frac{\pi}{4}$ My attempt : I simplified the integrand and got the below integral. $\int_{1}^{e^2}\frac{(lnx\...
Sparsh Gupta's user avatar
8 votes
4 answers
235 views

How to prove this integral inequality $\int_{0}^{1}[f''(x)]^2dx\ge 12$?

Let $f\in C^{2}[0,1]$ such that $f(0)=0,f(1/2)=f(1)=1$. Show that $$\int_{0}^{1}[f''(x)]^2dx\ge 12.$$ My idea: try to find a function $f(x)$ such that the equality holds and use Cauchy-Schwarz. So I ...
Schröchin's user avatar
1 vote
1 answer
67 views

Prove an integral inequality with squared integrals

Given $f, g$ integratable prove that $$\left(\int_0^1 f(t) \ \mathrm{d}t\right)^2 + \left(\int_0^1 g(t) \ \mathrm{d}t\right)^2 \leq \left(\int_0^1 \sqrt{f^2(t) + g^2(t)} \ \mathrm{d}t\right)^2$$ I ...
ABlack's user avatar
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0 votes
1 answer
26 views

Does this inequality for sums hold for integrals too?

Let $f,g:I \to \mathbb R$ be positive functions on a finite set, then $$ \left(\sum_i f(i)\right)\left(\sum_i g(i)\right) \geq \sum_i f(i)g(i) $$ The proof is rather trivial: by multiplying out the ...
seldon's user avatar
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1 vote
1 answer
80 views

Prove that: $(\int_{0}^{1/2}e^x(f(1-x) - f(x))dx)^2 \le (e-1)\int_{0}^{1}(f'(x))^2dx$.

Let $f:[0,1]\to\mathbb{R}$ a differentiable function, with its derivative $f'$ continuous, so that $f(0) = f(1)$. Prove that: $(\int_{0}^{1/2}e^x(f(1-x) - f(x))dx)^2 \le (e-1)\int_{0}^{1}(f'(x))^2dx$. ...
mathman's user avatar
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0 votes
1 answer
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Given concave decreasing function, does $\exists c\in [0,1]$ s.t. $\frac{(f(c)-cf'(c))\left(c-\frac{f(c)}{f'(c)}\right)}{2}\leq2\int_{0}^{1}f(x)dx?$

For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ ...
Adam Rubinson's user avatar
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0 answers
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Integral inequality on a special domain (Courant- Hilbert)

I am trying to understand a proof in [Maz’ya, 2011] Sobolev Spaces: With Applications to Elliptic Partial Differential Equations. Springer. Let $\Omega \subset \mathbb{R}^2$ be given as bellow (Figure ...
guerraufo's user avatar
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1 vote
1 answer
219 views

How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+...\frac{1}{n!}) = e$ with $x_n = (1+\frac{1}{n})^n$?

I'm very confused about the question below, which I couldn't figure out for days. In Example 5 the author is teaching us proving $$\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{...
John HHU's user avatar
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0 answers
29 views

Maximal mean value of a differentiable function with small variation

Let $B$ be a real valued function on $[0, 1]$ such that $\|B'\|_{L_2}\le 1$ and $B(0) = 0$. Denote \begin{gather}\label{small var condition} \varepsilon = \int_0^1 B^2(s)\, ds - \left(\int_0^1 B(s)...
Pavel Gubkin's user avatar
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0 answers
38 views

Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$ [duplicate]

I want to prove the inequality with analysis methods instead of method of PDE Actually we can consider the following PDE: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ ...
Gang men's user avatar
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0 answers
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Theorem 7.20 in Apostol's MATHEMATICAL ANALYSIS, 2nd edition: The Comparison Theorem for Riemann-Stieltjes Integrals

Here is Theorem 7.20, in Chap. 7, in the book Mathematical Analysis - A Modern Approach To Advanced Calculus by Tom M. Apostol, 2nd edition: Assume that $\alpha \nearrow$ on $[a, b]$. If $f \in R(\...
Saaqib Mahmood's user avatar
2 votes
1 answer
82 views

Prove that $\int f \ln(f) d \mu =\sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$

Question Prove that $\int f \ln(f) d \mu = \sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$ With $f$ verifying $ \int f d \mu = 1 $ and $ f \cdot \ln(f) $ is integrable, with $ \...
OffHakhol's user avatar
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0 answers
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Some integral inequality question with convexity

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a convex and integrable function such that $\lim_{x\to \infty} f(x) - x$ exists and is finite. I am also given the fact that $\int_{0}^1f(x)dx=\frac{1}{2}$. ...
FireWolf15G8's user avatar
-1 votes
1 answer
65 views

An integral inequality involving exponentials and $H^1(\mathbb{R}^2)$ [closed]

Let $\beta, \alpha>1$, and $u \in H^1(\mathbb{R}^2$). I'm trying to show that $$\int_{\mathbb{R}^2}\left(e^{\beta u^2(x)} -1\right)^{2\alpha}dx \leq \int_{\mathbb{R}^2}(e^{2\alpha \beta u^2(x)}-1)...
toothlessninjafrog's user avatar
3 votes
1 answer
76 views

$L^p_x$ norms of $f(x, x+y)$

Let $f \in \mathcal{S}(\mathbb{R}^2)$. Then do there exists inequalities of the form $$\lVert f(x, A(x,y))\rVert_{L^p_x(\mathbb{R})} \lesssim_{A, p,q} \lVert f(x,y) \rVert_{L^q(\mathbb{R}^2)} $$ where ...
newbie's user avatar
  • 307
3 votes
2 answers
128 views

convex function inequality similr to the Hermite-Hadamard inequality

Given a function $f: [a,b] \to \mathbb{R}$, that is convex and continuous, prove that the following inequality: $$\frac{2}{b-a}\int_a^b f(x) \,dx\le\frac{f(a)+f(b)}{2}+f\left(\frac{a+b}{2}\right).$$ I ...
FireWolf15G8's user avatar
2 votes
1 answer
48 views

Geometric-exponential series inequality

In this publication, in proof of theorem 5.11 there is transition which I interpret as inequality $$ \sum_{k=0}^\infty 2^{Nk} \exp\left(-\frac {4^k \rho^2} {Dt}\right) \leq C_{N, D} \exp\left(-\frac {\...
Esgeriath's user avatar
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0 votes
1 answer
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Limit representation of the Gamma function

I was going through the proof for the limit representation of the Gamma function where to prove the interchange of the limit and integral is justified, the author uses the following two relationships ...
Benjamin Kurian's user avatar
1 vote
0 answers
69 views

$L^{p}$ norm of $\frac{|\nabla f|^{2}}{f}$

I have asked this question in mathoverflow https://mathoverflow.net/questions/461885/lp-estimate-for-frac-nabla-f2f . I think I should also ask here. I’m trying to obtain an $L^{p}$ estimate under ...
Xin Qian's user avatar
1 vote
1 answer
133 views

Prove the following inequality given a differential inequality.

Let $y(t)' \leq 2ty(t)+\sqrt{y(t)}$ s.t. $y(0)=0$ where $y(t):[0,\infty) \to [0,\infty)$ continuous and differentiable on $(0,\infty)$. Prove the following $$y(t)\leq \frac{1}{4}\left(e^{t^2/2} \int_{...
Jam's user avatar
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1 vote
0 answers
71 views

Uniform bound for energy estimate

Consider the viscous Burgers equation on the torus $\mathbb{T} = \mathbb{R}/(2\pi\mathbb{Z})$, with $\nu > 0$ and $f$, $u_0$ in $L^{2}(\mathbb{T})$ which are assumed to be time-independent and mean-...
random's user avatar
  • 71
0 votes
2 answers
56 views

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
8 votes
1 answer
116 views

weakened $L^p$ interpolation using the Fourier transform?

It is easy to see that $$f\in L^\infty(\mathbb R^d), \ f\in L^1(\mathbb R^d) \implies f\in L^p(\mathbb R^d) \ \text{for all}\ p\in[1,\infty] \label{1}\tag{A}$$ Indeed, $ \int|f|^p \le \|f\|_{L^\...
Calvin Khor's user avatar
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2 votes
0 answers
48 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
0 votes
1 answer
42 views

Example that the Poincare inequality fails for higher exponent

I am trying to bound a function $u$ on a ball $B\subset \mathbb R^n$ using the norm of its gradient. I found in many literatures that $\lVert u-\bar u\rVert_{L^q(B)}$$\lesssim_ {p,q,n}\lVert\nabla u\...
flowing's user avatar
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2 votes
0 answers
98 views

How to estimate $\int_0^{L/2} f(x)dx$ from $f(x)\ge 2x-\frac{L}{D} \int_0^x f(t)dt$

$D>0, L>0$ are constant. $f:[0, +\infty) \rightarrow \mathbb R$ is non-negative continuous function. $f(0)=0$ and $D=\int_0^{+\infty} f(x) dx$. In fact, when $x>0$ is large enough (than $L$),...
Enhao Lan's user avatar
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0 answers
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Prove that $\left(\int_0^1 f(x)dx\right)^2\leq\frac{1}{12}\int_0^1\left(f'(x)\right)^2dx$ [duplicate]

The following problem is from Andreescu's "Problems in Real Analysis: Advanced Calculus on the real axis". More specifically it's problem $10.6.11$: Let $f:[0,1]\rightarrow\mathbb{R}$ be a ...
Summand's user avatar
  • 362
3 votes
0 answers
188 views

Integral inequality from AMM 1992

I would like to know the solution of the following 1992 AMM problem: Let $f$ be a continuous non-negative function defined on the square $[0,1]^2$. Show that $$ \int_0^1\int_0^1\int_0^1\int_0^1f(x_1,...
Pavel Gubkin's user avatar
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0 votes
0 answers
33 views

Comparing two different weighted averages of a function. [duplicate]

I need to prove the following: Assumptions: $f(\cdot)$ and $g(\cdot)$ are continuous and weakly increasing on $[0,1]$. $f(x)\geq 0$ and $g(x)\geq 0$ for $\forall x \in [0,1]$. $\int_0^1g(x)dx = 1$. ...
EconKR00's user avatar

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