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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How to prove Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have $$(f(t))^4\le \left(\int_{-1}^{1}\dfrac{[2(1-|x|)f'(x)-f(x)][2(1-|x|)f'(x)+f(x)]}{4(1-|x|^2)}dx\right)\cdot\left(\int_{-1}^{...
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Is it true that $\phi(\mu)=\mu F(\mu)^2-\int_{\mu}^{\overline{v}}F(v)[1-F(v)]dv\geq 0$?

Consider a random variable $V$ with distribution function $F$ and density function $f$ with support $[\underline{v},\overline{v}]$, where $0\leq\underline{v}<\overline{v}$. The mean is $\mu$. Here $...
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How to prove or falsify this inequality?

In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi $ $$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$...
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Are there any interpretations for the Gronwall's inequality in view of comparison theorem?

One form of the Gronwall's inequality is that If $\alpha(x),u(x)$ are non-negative continuous functions on $[0,1]$, and $$\forall x\in [0,1], u(x)\leq C+\int_{0}^{x}[\alpha(s)u(s)+K]ds\;(C,K\geq0),$$...
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Existence theorems for self adjoint elliptic systems

Let's consider an elliptic (vectorial, homogeneous, constant coefficient) system of order 2m $$ \begin{cases} Lu=f &\text{in }\Omega\\ Bu=0 &\text{on }\partial\Omega \end{cases} $$ which is ...
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Convolution by $\log$ maps $\mathrm{L}^1$ into $\mathrm{BMO}$

It is stated in Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Ch. IV, 6.3(i) that $$ I_{n} f:=f\star\log|\cdot|\in\mathrm{BMO}(\mathbb{R}^n)\qquad\text{if ...
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Probs. 10 (a), (b), and (c), Chap. 6, in Baby Rudin: Holder's Inequality for Integrals

Here is Prob. 10, Chap. 6, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $p$ and $q$ be positive real numbers such that $$ \frac{1}{p} + \frac{1}{q} =1. $$ Prove ...
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Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
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A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
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Minkowski's integral inequality for other norms

In my measure theory course we studied norms $L_p$ and no other norms. For proofs we used exclusively the trick known as Hoelder's inequality which works only on $L_p$ norms. I disliked it very much ...
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For cubic $P(x)$ with a root in $[0,1]$, find the smallest $C$ such that $\int_0^1 |P(x)|dx\leq C\max_{x\in[0,1]}|P(x)|$

Find the smallest constant $C$ such that for every polynomial $P(x)$ of degree $3$ with a root in $[0,1]$, $$\int_0^1 |P(x)|dx\leq C\max_{x\in[0,1]}|P(x)|.$$ Here's my rough work. Write $P(x)=a_3x^3+...
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How did he use Gronwall Lemma??

I´ve got these lines from an article: ( where $b:\mathbb{R}_+\to \mathbb{R}_+$ is non-decreasing and $(X_t)$ is an $\mathbb{R}_+$-valued process - it doesn't matter very much, I guess, anyway-.) $$...
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hard integral inequality with $e^{x^2}$

a) prove the convergency of $$ \int_0^{\infty} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}$$ b) prove the inequality $$ \int_0^{\frac {7\pi} {12}} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}<...
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Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
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$L^p_x$ 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 ...
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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,...
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Prove $\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$ for reals $\alpha_1,\alpha_2,...\alpha_n$

Prove for all positive integer $n$ and for all $\alpha_1,\alpha_2,...\alpha_n \in \mathbb{R}$ Prove that $\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$ I have tried by applying ...
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Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.

Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
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Energy bound for a closed curve

Let $\gamma : S^1 \rightarrow M$ be a smooth map from a circle of length 1 to a closed manifold $M$ with nonpositive curvature. Could we find a constant $C > 0$ depending only on $M$ such that $$\...
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Inequality with integral and distribution function

I encountered an inequality with the following variables and functions. $X$ is a random variable drawn from $(-\infty, \infty)$ with cdf $F$, pdf $f$, and mean $\mu=\mathbb{E}[X]$. For any $x$ and $\...
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How to prove this integral inequality by variational method?

Suppose $f(x)$ is fourth differentiable and satisfies $f(0)=f(1)=f'(0)=f'(1)=0$.Prove the inequality$$\frac{1}{a^4}\int^{1}_{0}[f''(x)]^2\text dx\geq \int^{1}_{0}[f(x)]^2\text dx$$Where $a$ is the ...
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Prove integral is convex

$X$ are an iid draw from $(-\infty, \infty)$ according to $F$ with mean $\mu$. Further let $A = a(x, \theta)/\cos (\alpha)$ and $B = ((1- \cos(\alpha) - \sin (\alpha))\mu + \sin (\alpha)x)/\cos(\...
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Estimating $\int_0^{2\pi}\left(\sum_{n=1}^{N} \cos (n^2x) \right)^4\ dx$ for large $N$

In a paper I am reading, it is claimed without proof that for every $\varepsilon > 0$ and $N > N_0(\varepsilon)$ $$ \int_0^{2\pi}\left(\sum_{n=1}^{N} \cos (n^2x) \right)^4\ dx < N^{2 + \...
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Non-existence of supremum norm bound in terms of $L^2$ norm of gradient

Show there does not exist any constant $C>0$ such that, for any $\phi\in C^{\infty}_c(\mathbb{R}^2)$, the inequality $$\lVert\phi\rVert_\infty\leq C\lVert\nabla\phi\rVert_2$$ holds. Obviously, we ...
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Two-index Hajek-Renyi type inequality

Consider a sequence of mean-zero independent random variables $\{X_i\}_{i=1}^\infty$ and the partial sum $S(n)=\sum_{i=1}^n X_i$. The Hajek-Renyi inequality provides a upper bound of the normed ...
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A singular integral

So, I was doing some PDE related computations and I obtained the following integral $$ \iint \frac{(y-y')\,f(x',y')}{(|x-x'|^2+|y-y'|^2)^\frac{3}{2}}\, \left|\frac{x'}{x}\right|^2 \,\mathrm{d} x'\,\...
LL 3.14's user avatar
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Immersion $H^s(\Omega) \hookrightarrow H^{s'}(\Omega)$ but with fractional laplacian defined on $\mathbb{R}^N$?

I know that it holds this embeding $$H^{s}(\Omega) \hookrightarrow H^{s'}(\Omega)$$ for $s>s'$ and for any $\Omega \subset \mathbb{R}^N$. In this case, anyway, the fractional laplacian is defined ...
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Minimizing $\int_0^1 [f''(x)]^2 dx$ with constraints

I would like to find the greatest lower bound and hopefully the function that minimizes $\int_0^1 [f’’(x)]^2 dx$ where $f\in C^2([0,1])$ with $f(0) = f(1) = 0$ and $f’(0) = 1$. The only thing I could ...
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Burkholder-David-Gundy ineq. with conditional expectation?

Let $\sigma$ be a progressively measurable stochastic process on a probability space with a standard Brownian motion $W$. The Burkholder-David-Gundy inequality tells us that $$ \mathbb E \Bigg( \sup_{...
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Examples of Besov functions of power-logarithmic type $|x|^{\alpha} |\log |x||^{\beta}$

I'm stuck on the following exercise 17.9 from Leoni's text A first course in Sobolev Spaces (second edition). This is from the chapter on Besov spaces, but this is really just a integral inequality ...
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Is it a well-known inequality?

Let $f\in \mathcal{C}^2(\mathbb{R},\mathbb{R})$ and suppose that $\int_\mathbb{R}f^2<+\infty$ and $\int_\mathbb{R}f''^2<+\infty$. Then we can deduce that : $\left(\int_{\mathbb{R}}f'^2\right)...
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A Gronwall type inequality involving iterated integrals

Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have: $$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$ where $\kappa, C >...
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How to prove the following ineqiulity : $\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,s)f(s)}ds$

Let $f$ be integrable on $[0, t]$ , $t≥0$ then prove that $$\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,t)f(s)}ds$$ This clearly smells like Jensen's inequality where we instead ...
Guy Fsone's user avatar
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3 votes
1 answer
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Seeking a Proof or a counter example for the following step regarding inequalties

Please consider if the following pair of inequalities hold $$ g(X) \le \mathbb{P}\left(\xi > X \right) \le h(X) $$ where the domain of integration is over some variable $x$ and $X$ is some constant ...
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3 votes
0 answers
308 views

Inequality with Products of Integrals

Following from a previous post: Let $h_i, h_j, f_i,f_j$ continuous functions such that $f$ are increasing and $f(0)=0$, Assume that: $\forall x$ $$h_i (x)>h_j (x)$$ $$f_i (x)>f_j (x)$$ and $$...
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Minkowski inequality for integrals

Let $(X_1,\mu_1)$ and $(X_2,\mu_2)$ be two $\sigma$-finite measure spaces. Let $f(x_1,x_2)$ be a measurable complex valued function. Stein and Shakarchi's 4th book asks to prove that $$ \|{\int f(x_1,...
Manan's user avatar
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3 votes
1 answer
496 views

Fractional integral inequality (Hardy-Littlewood)

I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
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3 votes
0 answers
200 views

Is this function increasing?

I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$: $$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
Emmanuel's user avatar
3 votes
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53 views

Inequality involving homogenous function of degree -1

Let $p\in[1,\infty]$. For $f\in L^p(0,\infty)$ we define $Tf:x\mapsto \int_0^\infty K(x,y)f(y)\,dy$ where $K$ is homogenous of degree $-1$, i.e. $K(\lambda x,\lambda y) = \lambda^{-1} K(x,y)$ for $\...
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3 votes
1 answer
389 views

Using Gronwall's Inequality with Random Variables

Currently, I've been working with an SDE and trying to get a bound on moments. I have it down to something of the following form: $$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$ ...
Brenton's user avatar
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Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and $\frac{1}{p}+\frac{1}{q}=\frac{...
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3 votes
0 answers
121 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to [0,\...
Fin8ish's user avatar
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An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
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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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94 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 $L$),...
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Inverse inequality

I would like to prove the following inequality: Let $f$ be a function $f:\mathbb R\to\mathbb R$ such that $0\leq f\leq 1$ on $\mathbb R$ and $f=1$ on $[1,+\infty)$ and $f=0$ on $(-\infty,-1]$. Prove ...
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Understanding Proof of Talagrand's Inequality

I am reading Talagrand's seminal paper Concentration of Measure and Isoperimetric Inequalities in Product Spaces. Lemma 2.1.2 on Page 12 obtains the bound $$ \int_\...
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What is the correct version of the Gronwall lemma? Can the sign of u(t) be variable?

In https://encyclopediaofmath.org/wiki/Gronwall_lemma the various forms of the Grönwall lemma in integral form are stated for NON NEGATIVE function $\phi$ And this coincides with what is written my ...
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Rudin RCA Problem 4.12 (Hint Request)

Clarification and Attempted Solution I've been stuck on this problem for several days now, and I'm entirely frustrated at this point. I do not know how to estimate the integral in the particular way ...
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For finite-duration continuous $f(t)$ with $\|f'(t)\|_\infty < \infty$: It is true $\|f'(t)\|_\infty \leq \frac{2\pi \|f'(t)\|_2^2}{\|f'(t)\|_1}$?

For finite-duration continuous $f(t)$ with bounded derivative $\|f'(t)\|_\infty < \infty$: It is true that $\|f'(t)\|_\infty \leq \frac{2\pi \|f'(t)\|_2^2}{\|f'(t)\|_1}$? I am looking for an upper ...
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