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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Integral Inequality involving a binomial

Let $\binom{t}{k}$ where $k\in \mathbb{N}$ be defined as $\frac{t(t-1)\cdots (t-k+1)}{k!}$. Prove that $\int_n^{\infty} \binom{t-1}{n-1} e^{-t} dt \le \frac{1}{(e-1)^n}$ My progress is as follows: we ...
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  • 402
2 votes
1 answer
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Show that $\displaystyle\sum_{i=1}^k\pi_i\int_{R_i} f_i(x)dx\ge\sum_{i=1}^k\pi_i^2$

I've got a problem in Statistical Inference which I've reduced to showing $\displaystyle\sum_{i=1}^k\pi_i\int_{R_i} f_i(x)dx\ge\sum_{i=1}^k\pi_i^2$ Here $\pi_i>0\:\forall\:1\le i\le k,$ and $\sum_{...
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A integral inequality in the paper "The concentration-compactness principle".

In the paper <The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1.> by P. L. Lions, I am confused with one inequality (Page 125, Line 8 from ...
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4 votes
1 answer
173 views

proof verification for an integral inequality

Problem: Let $f(x)$ be a continuous function on $[a,b]$, such that $|f(x)|\leq M$ and $\int_a^b f(x)dx = 0$. Prove that $$|\int_a^b xf(x)dx| \leq \frac{M}{4}(b-a)^2.$$ Can anyone help verify my proof? ...
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Is there a reference for this well-known inequality?

Here is the statement : Let $f \in \mathcal{C}^1([0,T],\mathbb{R}_+)$, $T>0$ such that : for all $t \in [0,T]$, $\displaystyle f(t) \le c + \int_{0}^{t} \eta(f(x))\mathrm{d}x$ where $\eta : \...
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2 votes
1 answer
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Does this version of Grönwall's lemma hold?

Here is the lemma with more general hypothesis : Let $f,g,y\in L_{\text{loc}}^1(\mathbb{R})$ non-negatives such that for all $t \in \mathbb{R}$, $\displaystyle y(t) \le f(t) + \left \vert \int_{t_0}^...
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  • 3,130
2 votes
1 answer
72 views

Find the maximum value of $\int_0^1 (f(x))^3 dx$, under certain conditions on $f(x)$ and $\int_0^1f(x)dx$

Problem I have gone through the solution here Finding the maximum value of $\int_0^1 (f(x))^3 dx$, given certain conditions on $f(x)$ and $\int_0^1 f(x) dx$ and satisfied with it but I am unable to ...
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1 answer
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Simplifying Integral Inequality with e on one side and ln on the other

Can you help me finish this problem? Decimal approximations are not sufficient for credit. I wasn't sure how to simplify the left side much more (besides obviously multiplying by two and subtracting ...
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1 vote
1 answer
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Proof verification: an integral inequality

I wrote down a proof for the following proposition, and it is quite different from the standard proof which views the integral as a function of upper limit. Can anyone please verify it for me? ...
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An inequality for a function of two variables

Suppose $x,y\in [0,\pi]$, does the inequality $$f(x,y) = x+y-x\,e^{\sin^2x-\sin^2y}-y\,e^{\sin^2y-\sin^2x} \leq 0$$ always hold? I have been trying to prove the following integral inequality $$\int_0^{...
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3 votes
1 answer
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A simple special case of Gronwall's inequality for Dini derivatives

Let $I=[t_0,t_1)\subset \Bbb R$ an interval and $a,b,c\ge0$ with $a>c$. Assume that $f\colon I\to\Bbb R$ is a continuous function with $$\tag{1} f(t)-f(s)\le \int_s^t\left( -af(r)+be^{-cr} \right) ...
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3 votes
3 answers
107 views

Functional inequality with integral function [closed]

Given a function $f:[0;1]\to[0;1]$ such that $f(x)\leq2\int_0^x f(t)dt$, prove that $f(x)=0$ $ \forall x\in [0;1]$. I've observed that the function has to be concave down in his domain and that $...
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1 vote
2 answers
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Show that $\int_2^{+ \infty} \frac{\log(t)^2}{t(t-1)}dt \leq 4$

I couldn't prove this inequality $$ \int_2^{+ \infty} \frac{\log(t)^2}{t(t-1)}dt \leq 4 $$ I've tried integration by parts but it doesn't work.
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2 votes
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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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1 vote
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Integration over the intersection of a hypersphere and half spaces

For $x\in\mathbb{R}^n$, $u\in\mathbb{R}^n$ and $0<c<\Vert u\Vert$, how could I compute (or find the upper bound of) the following integral $$\int_{x^\top x =1,~\vert u^\top x\vert\leq c}1dx.$$ ...
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Derivation of Hölder Inequality through Young's Inequaliy

I am having trouble following a proof where Young's Inequality is being used to derive Hölder's Inequality. More precisely, there is a particular and final step that utilizes integration in order to ...
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1 vote
1 answer
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How to evaluate the following integral might be related to the modified Bessel Function of first kind?

Recently, I have encountered the following integral solution problem in my research. Because it involves special functions, I cannot successfully solve it in calculation. $$\mathbb{E}_{Z_{1},Z_{2},\...
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How to solve the following double integral with the Bessel function?

Recently, I have encountered the following integral solution problem in my research. Because it involves special functions, I cannot successfully solve it in calculation. $$\mathbb{E}_{Z_{1},Z_{2}} \...
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  • 53
1 vote
1 answer
94 views

Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$

The number of continuous functions $f:\left[0,\frac{3}{2}\right]\rightarrow (0,\infty)$ satisfying the equation$$4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$$ ...
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1 vote
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Showing inequality for the norm of an integral equation

Let $\xi \in \mathbb{R}^2$, $\Phi \in C^0([0, +\infty[, \mathbb{R}^{2x2})$ a bounded function. Let $y:[0, +\infty[ \rightarrow \mathbb{R}^2$ a solution of the following integral equation, $$ y(x) = e^{...
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  • 577
2 votes
1 answer
39 views

Is this kind of inverse substitution justified?

I'm trying to prove that for every integer $n\geq 0$ we have $$\int_0^{\pi/2} (1+\cos t)^ndt\geq \frac{2^{n+1}-1}{n+1}.$$ I started out by rewriting the RHS as $\int_0^{1}(1+x)^ndx$ and substituting $...
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2 votes
2 answers
30 views

Inequality involving moments of a distribution [closed]

Let X be a real random variable. Under what conditions on the distribution do we have that $$\mathbb{E}( X^{2n + 2}) \geq \mathbb{E}( X^{2n}) \mathbb{E}( X^{2})$$ for all integer $n$? I tried using ...
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  • 29
6 votes
1 answer
100 views

What functions satisfy $\int_a^b f(x) c(x) \, dx \ge 0$ for all convex functions $f$?

This is an attempt to generalize Prove that $\int_{0}^{2\pi}f(x)\cos(kx)dx \geq 0$ for every $k \geq 1$ given that $f$ is convex. Inspired by that question and the given answers, I have the ...
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an integral inequality from "Inequality" by Hardy,Littlewood and Polya Chapter7

If $f\in C^1[0,1)$ and $f(0)=0$ , show that $\int_0^1\frac{|f(x)|^2}{x^2}dx\leq4\int_0^1|f^{'}(x)|^2dx$ The book solves the problem using variational method. But I want to seek for an elementary ...
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3 votes
2 answers
67 views

Proving two inequalities involving integrals and sums

Let $p \ge 1$. Show that \begin{equation} \frac{1}{(k+1)^p} \le \int_{k}^{k+1} \frac{1}{x^p} \,dx \le \frac{1}{k^p} \end{equation} for every positive integer $k$. Hence show that \begin{equation} \...
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1 vote
0 answers
27 views

Fokker Planck Equation with strongly convex potential function

Let $\rho_t$ be the (weak) solution of Fokker-Planck equation $$ \partial_t\rho_t = \nabla\cdot(\rho_t\nabla(\log\rho_t + V)) $$ with initial condition $\rho_0$ and no-flux boundary condition $\rho_t(\...
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3 votes
2 answers
122 views

Can I prove this inequality only by elementary tools?

Let $\Omega\in\mathbb{R}^2$ be a bounded domain, $\forall a,b,c \in \mathbb{R}$, I hope to prove that there exists a positive constant $\alpha$ such that: $$ \int_{\Omega} (ax_2+b)^2 + ( -ax_1+c)^2 \...
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  • 303
2 votes
3 answers
142 views

Prove that $x/((e+x)\ln(e+x)) \leq \ln(\ln(e+x)) \leq x/e$ for all $x > 0$

I've been working on a problem but have been stuck for several hours finishing it. The problem is to show that $$ \frac{x}{(e+x) \ln(e+x)} \leq \ln(\ln(e+x)) \leq \frac{x}{e} $$ for all $x > 0$. ...
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9 votes
1 answer
147 views

A fractional Gronwall lemma

I have an energy estimate of the form $$\dot{u} + \lambda u^{1+ \delta} \le C (1+t)^{-\sigma},$$ where $u=u(t)$ is positive, $\lambda, \delta > 0$ and one can assume $\sigma > 1$ large. I expect ...
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  • 221
0 votes
1 answer
76 views

An estimation about polynomials in the complex plane [duplicate]

I've been learning complex analysis recently, there's a question about the estimation of polynomials on the real line using the knowledge of complex analysis. Suppose $f(x)={c_0}+{c_1}x+{c_2}x^2+...+{...
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  • 21
0 votes
1 answer
34 views

HLS inequality not suitable to bound this integral

I am trying to bound the following integral for $f \in L^{n/2}(\mathbb{R}^n)$: $\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} f(x) \lvert x-y \lvert^{2(2-n)}f(y)dxdy$. Because of the factor 2 in the exponent,...
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  • 329
1 vote
1 answer
33 views

The seminorm of $L^{p,\infty}$ does not fulfill the triangular inequality.

Hello I am attending a course of harmonic analysis. My professor pointed out a fact about $L^{p,\infty}(\mathbb{R^{d}})$ spaces, leaving the proof to us, and I got stuck. It is not an assignment of ...
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5 votes
1 answer
88 views

Example for a continuous function $x \geq 0$ on $[0,\infty)$ so that $x(0)=0$ and $\left(x(t) \right)^2\leq 2+\int_{0}^{t}x(u)du,~~~\forall ~t\geq 0$

Q. Suppose $x:[0,\infty)\to [0,\infty)$ is continuous and $x(0)=0.$ If $$\left(x(t) \right)^2\leq 2+\int_{0}^{t}x(u)du,~~~\forall ~t\geq 0,$$ then which of the following is TRUE? $x(\sqrt{2})\in [0,2]...
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10 votes
2 answers
247 views

An apparently simple but somehow unexpected inequality between integrals

Let $f:[0,1]\to[0,1]$ be any Lebesgue integrable function. Then $$I_1= \left(\int_0^1f(x)\sqrt x dx\right)^2 \leq \frac{4}{3}\int_0^1f(x)x^2dx=I_2\,.\tag{$\star$}$$ At first sight, I would have ...
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  • 2,820
0 votes
0 answers
18 views

Finding an Integral Lower Bound for a Weierstrass-like Function.

Here is a problem I have been stuck on for quite a while. Let $X = (-\pi, \pi)$, $0 < s < 1$ and $0 \leq \delta \leq \frac{1}{2}$. We define the function $$ f(x) := \sum_{n = 1} ^\infty \frac{1}{...
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0 votes
0 answers
18 views

Integral decay implying pointwise decay

Let $\varphi : [1,\infty) \to [1,\infty)$ be a non-decreasing function such that $$ \int_1^{\infty} \frac{\mathrm{d}t}{\varphi(t)} = + \infty. $$ Then does there exist $p>1$ and $C>0$ such that $...
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2 votes
0 answers
53 views

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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  • 164
6 votes
1 answer
123 views

Using Mantel's theorem to prove a probabilistic inequality

I'm self-learning Yufei Zhao's "Graph Theorey and Additive Combinatorics", one problem I encounter is the following: The next exercise can be solved by a neat application of Mantel's ...
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3 votes
0 answers
163 views

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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  • 1,157
3 votes
2 answers
99 views

Convexity of the exponential of the negative Renyi entropy

For $r\ge -1$, the exponential of the negative Renyi entropy is defined as $$M(p):=\Big(\sum_i p_i^{1+r}\Big)^{\frac1r},$$ for a probability measure as tuples $p:=(p_i)_i$ I would like to prove the ...
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  • 9,102
0 votes
0 answers
52 views

Removing the square roots of Integrands to compare two integrals

I am trying to prove an inequality concerning two integrals as part of proving that an extremum in a Calculus of Variations problem related to special relativity is a global minimum. I started with ...
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  • 721
0 votes
1 answer
42 views

Integral inequality for positive functions [closed]

Is it true that $ \int_Sf(x)g(x)dx \leq \int_Sf(x)dx\int_Sg(x)dx \ \forall f(x),g(x)\geq0 $ ?
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0 votes
0 answers
58 views

Reversed Cauchy-Schwarz inequality of improper integral

I'm wondering for given functions $f$ and $g$, such that $0\leq f,g < \infty$ on $(0,1]$ $\lim_{x\to0} f(x) = \infty$, and $\lim_{x\to0}g(x) < \infty$ $f^2$ and $g^2$ are (Riemann) integrable ...
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1 vote
1 answer
102 views

Maximum of a Function Bounded by Average of Integral and Integral of Derivative [duplicate]

Post Taken Down Post Taken Down Post Taken Down Post Taken Down Post Taken Down
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2 votes
0 answers
58 views

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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  • 1,031
0 votes
1 answer
45 views

Does $||f'||_\infty \leq \sqrt{t_F-t_0}\,||f'||_2$ hold for time-limited continuous functions $f(t)$ with $\sup_t |f'(t)|<\infty$?

Prerequisites for the answer: I am trying to understand which conditions makes the derivative of non-everywhere-differentiable continuous time-limited functions to be bounded $\sup_t |f'(t)|<\infty$...
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  • 1,031
0 votes
1 answer
57 views

Show $\int_{0}^{1}(\int_{0}^{x}g(t)dt)^2 dx\leq\frac{1}{2}\int_0^1(1-x^2)(g(x))^2 dx$ for any $g(x)$ continuous

Prove (or disprove) that $$\int_{0}^{1}\left(\int_0^x g(t)\ dt\right)^2dx\leq\frac{1}{2}\int_0^1 (1-x^2)(g(x))^2 dx$$ for any $g(x)$ continuous on $[0,1]$. I have verified the cases of $g(x)$ being ...
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  • 37
1 vote
1 answer
62 views

Which conditions must fulfill $f(t)$ to have an absolute-integrable Fourier Transform $F(w)$: $\int\limits_{-\infty}^\infty |F(w)| dw < \infty$?

Which conditions must fulfill $f(t)$ to have an absolute-integrable Fourier Transform $F(w)$: $\int\limits_{-\infty}^\infty |F(w)| dw < \infty$? At first, thinking in $f(t)$ as an arbitrary one ...
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  • 1,031
1 vote
1 answer
95 views

How to prove $\int_0^1 f^2(x)\cdot f'^4(x)\ dx\leq \int_0^1 f^4(x)\cdot f''^2(x)\ dx$ [closed]

Suppose $f(x) \in C^2[0,1]$ with $f(x)>0\quad\forall x\in [0,1]$. If $f'(0)=f'(1)=0$,¿ how to prove $$\int_0^1 f^2(x)\cdot (f')^4(x)\ dx\leq \int_0^1 f^4(x)\cdot (f'')^2(x)\ dx$$
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  • 155
0 votes
1 answer
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$L^2$ norm and $L^{\infty}$ inequality for periodic smooth functions

Let $\varphi \in C^{\infty}(\mathbb{T}^n,\mathbb{C})$ (i.e. just smooth periodic complex-valued function) and $f \in C^{\infty}(\mathbb{T}^n,\mathbb{C}^m)$. Then I was wondering if the following is ...
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