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

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18
votes
1answer
651 views

Do inequalities that hold for infinite sums hold for integrals too?

Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
16
votes
1answer
292 views

Prove the following integral inequality

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: ...
14
votes
4answers
1k views

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$ $(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
13
votes
2answers
534 views

Proof of bound on $\int t\,f(t)\ dt$ given well-behaved $f$

I got the following question by mail from someone I don't know from Adam. (Quoted in part.) if $f(t)$ continuously diff. on $[0,1]$ and a) $\int_0^1f(t)\ dt=0$ b) $m\le f\,'\le M$ on ...
12
votes
2answers
159 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
11
votes
6answers
392 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
10
votes
2answers
362 views

How prove this integral inequality $\int_{0}^{1}f^2(x)dx\ge 24\left(\int_{0}^{1}f(x)dx\right)^2$?

let $f:[0,1]\longrightarrow R $ be a continuous function, if $$\int_{0}^{1}x^2f(x)dx=-2\int_{\frac{1}{2}}^{1}F(x)dx$$ where $F(x)=\displaystyle\int_{0}^{x}f(t)dt,x\in [0,1]$,then prove that ...
10
votes
1answer
204 views

Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that $$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$ Prove that $$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$ Where should ...
9
votes
4answers
2k views

Jensen's inequality for integrals

What nice ways do you know in order to prove Jensen's inequality for integrals? I'm looking for some various approaching ways. Supposing that $\varphi$ is a convex function on the real line and $g$ is ...
9
votes
2answers
564 views

How to prove that $\frac{2}{\pi}\int_{x}^{px}\left(\frac{\sin{t}}{t}\right)^2\,\mathrm dt\le 1-\dfrac{1}{p}$ for $p >1, x\ge0$

Show that for $p>1$ and $x \ge 0$: $$\dfrac{2}{\pi}\int_{x}^{px}\left(\dfrac{\sin{t}}{t}\right)^2\,\mathrm dt\le 1-\dfrac{1}{p}$$ My idea is to use ...
9
votes
2answers
496 views

How prove this integral inequality $\int_{0}^{s}f(x)\,dx\le\int_{s}^{1}f(x)\,dx\le\dfrac{s}{1-s}\int_{0}^{s}f(x)\,dx$

let $f(x)>0$ is continuous and is increasing on $[0,1]$,and $s=\dfrac{\int_{0}^{1}xf(x)dx}{\int_{0}^{1}f(x)\,dx}$ show that ...
8
votes
6answers
578 views

How prove this $\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx$

let $f\in C^{(1)}[a,b]$,and such that $f(a)=f(b)=0$, show that $$\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx\cdots\cdots (1)$$ My try: use Cauchy-Schwarz inequality we have ...
8
votes
3answers
379 views

Inductively prove that this sequence of integrals is bounded.

EDIT: I have an attempted solution to this in a post below, it is very long, but still incomplete. EDIT:Alright, I've pretty much almost finished my solution, but my biggest problem is the 2nd ...
8
votes
3answers
471 views

How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$

let $f$ on $[a,b]$ two continuously differentiable functions,such $$f(a)=f(b)=0, f'(a)=1,f'(b)=0,b>a>0$$ show that $$\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$$ My idea: use ...
8
votes
3answers
137 views

An integral inequality with inverse

Let $f:[0,1]\to [0,1]$ be a non-decreasing concave function, such that $f(0)=0,f(1)=1$. Prove or disprove that : $$ \int_{0}^{1}(f(x)f^{-1}(x))^2\,\mathrm{d}x\ge \frac{1}{12}$$ A friend posed this to ...
8
votes
2answers
194 views

How prove this $\int_{-\pi}^{+\pi}\cos{(2x)}\cos{(3x)}\cos{(4x)}\cdots\cos{(2005x)}dx>0$

show that $$I=\int_{-\pi}^{+\pi}\cos{(2x)}\cos{(3x)}\cos{(4x)}\cdots\cos{(2005x)}dx>0$$ This problem is my frend ask me, My try: ...
8
votes
1answer
421 views

How to prove that there exists $g(x)$ such $\int_{0}^{1}g(x)dx\ge\frac{1}{2}\int_{0}^{1}f(x)dx$

let $f(x)\ge 0,x\in [0,1]$, and is increasing in $[0,1]$ show that: There exists $g(x)\ge 0,x\in [0,1]$,and $g''(x)>0$, such $g(x)\le f(x)$, and such ...
8
votes
1answer
138 views

A continuous function integral inequality

Let $m$ be a positive integer. $f\colon[0,\infty)\to[0,\infty)$ is a continuous function such that $f(f(x))=x^m,\forall x\in[0,\infty)$. Show that $$\int_0^1f^2(x)\,dx\ge\frac{2m-1}{m^2+6m-3}$$
8
votes
1answer
230 views

Holder's inequality for infinite products

In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of ...
7
votes
2answers
149 views

How prove this inequality $ 1-\cos (xy) \le\int_0^xf(t) \sin {(tf(t))}dt + \int_0^y f^{-1}(t) \sin{(tf^{-1}(t))} dt .$

Question: Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: $$ 1-\cos (xy) \le\int_0^xf(t) ...
7
votes
1answer
215 views

Show inequality of integrals (cauchy-schwarz??)

$f:[0,1]\to\mathbb{C}$ continuous and differentiable and $f(0)=f(1)=0$. Show that $$ \left |\int_{0}^{1}f(x)dx \right |^2\leq\frac{1}{12}\int_{0}^{1} \left |f'(x)\right|^2dx $$ Well I know that $$ ...
7
votes
2answers
452 views

An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
7
votes
1answer
65 views

How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$

let $$f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$$ show that: $$e^x|f(x)|\le 2$$ My idea: let $$e^t=u$$ then $$|f(x)|=|\int_{e^x}^{e^{x+1}}\dfrac{1}{u}d\cos{u}|$$
7
votes
1answer
175 views

Integral Inequality $|\int_0^x f(t)dt|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|$

Let $f\in C^2(\mathbb{R})$ such that $f(1)=\int_0^1f(x)dx=0$. Prove that $$\left|\int_0^x f(t)dt\right|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|\, \forall x\in [0,1].$$ Thanks in advance!
6
votes
2answers
135 views

How prove this $\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$

Show that this intergral inequality $$\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$$ I know this use Taylor's formula.But I think is very ugly,maybe this problem have simple methods.Thank you ...
6
votes
3answers
114 views

How prove this inequality $\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}\sin^n{x}dx\ge 0$

show that: $$\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}\sin^n{x}dx\ge 0$$ My idea:maybe $\sin{(x^n)}\ge (\sin{x})^n?$ Thank you
6
votes
2answers
276 views

Prove that if f(x) is integrable, then so is e^(f(x)).

So here is my question: I'm working on a homework problem that deals with Jensen's Inequality. It is a rather simple application, I believe, but I'm a little stuck. Here is the problem, along with ...
6
votes
1answer
194 views

How prove this integral inequality

let $f:[0,1]\longrightarrow R$ be a differentiable function with continuous derivative such that $f(1)=0$,show that: ...
6
votes
1answer
50 views

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
6
votes
1answer
846 views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...
5
votes
2answers
209 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
5
votes
1answer
91 views

Determining the best possible constant $k$, for an Integral Inequality

If $f : [0,\infty) \to [0,\infty)$ is an integrable function, then what is the best possible constant $k$, for which the following ineqality holds: $$\int_0^{\infty}f(x)dx \leq ...
5
votes
1answer
107 views

How show that $\dfrac{a^3}{3}\ge\int_{0}^{a}|F(x)-x|^2dx$

Let $F(x)$ be nonnegative and integrable on $[0,a]$ and such that $$\left(\int_{0}^{t}F(x)dx\right)^2\ge\int_{0}^{t}F^3(x)dx$$ for every $t$ in $[0,a]$,prove or disprove the conjecture: ...
5
votes
1answer
95 views

How prove this integral inequality $6\left(\int_{0}^{1}f(x)dx\right)^2\le 1+ 8\int_{0}^{1}f^3(x)dx$

Let $f$ be a positive-valued,concave function on $[0,1]$,Prove that $$6\left(\int_{0}^{1}f(x)dx\right)^2\le 1+ 8\int_{0}^{1}f^3(x)dx$$ Let ...
5
votes
1answer
98 views

Integral inequality $\int_0^{+\infty}|\frac{\sin x}x|^p dx\leq\frac\pi{\sqrt{2p}}$

$p\geq2$, then we have $$\int_0^{+\infty}\Bigg|\frac{\sin x}x\Bigg|^p\,\mathrm dx\leq\frac\pi{\sqrt{2p}}$$ I try to use $\Bigg|\frac{\sin x}x\Bigg|\leq1$, and $\frac{\sin ...
5
votes
0answers
38 views

If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...
5
votes
1answer
374 views

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 ...
4
votes
4answers
401 views

Proof of Schur's test via Young's inequality

I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality: Let ...
4
votes
1answer
98 views

Integral of composition [duplicate]

Prove that if $f,g:[0,1]\rightarrow[0,1]$ - continuous functions and f is strictly increasing then $$\int\limits_0^1f(g(x))dx\leq\int\limits_0^1f(x)dx+\int\limits_0^1g(x)dx.$$ I tried to prove that ...
4
votes
2answers
283 views

$3\int_{0}^{1}(f'(x))^2dx \geq (2\int_{0}^{1}f(x)dx)^2 \impliedby 2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$

Let $f : \mathbb{R} \to \mathbb{R} $ be a differentiable function. Suppose that $2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$ Show that $$3\int_{0}^{1}(f'(x))^2 ...
4
votes
2answers
223 views

How prove this integral inequality $2\int_{-1}^{1}f(x)g(x)dx\ge\int_{-1}^{1}f(x)dx\int_{-1}^{1}g(x)dx$

Let $f:[-1,1]\longrightarrow \mathbb{R}$ be increasing on $[0,1]$ and even, i.e. $f(x)=f(-x)$ $\forall x\in [-1,1]$. Let $g:[-1,1]\longrightarrow \mathbb{R}$ be convex, i.e. $g(tx+(1-t)y)\le ...
4
votes
1answer
112 views

Integral inequality with a function twice differentiable

Let $f:[0,1]\longrightarrow\mathbb{R}$ be a function twice differentiable with continous second derivative and $f(1)=f(0)$. The inequality: $$\int_{0}^{1}(f''(x))^2dx\geq ...
4
votes
1answer
73 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...
4
votes
1answer
187 views

Inequality in Schwartz space

I am trying to prove theorem 9.2 from book "Lectures on Linear Partial Differential Equations" wtitten by G. Eskin. In proof of this theorem is inequality which makes problems for me. Firstly I remind ...
4
votes
0answers
188 views

How prove $\frac{\int_{0}^{1}xf^2(x)dx}{\int_{0}^{1}xf(x)dx}\le\frac{\int_{0}^{1}f^2(x)dx}{\int_{0}^{1}f(x)dx}$ [duplicate]

let $f(x)$be positive and decreasing on $[0,1]$ show that: ...
3
votes
4answers
147 views

Integral inequality for continuous function

Let $ f $ be a continuous, real-valued function on $[0, 1] $. Show that $$\int_0^1 \int_0^1 |f (x)+f (y)| dx dy \ge \int_0^1 |f (x)| dx $$ I tried to dissect the square in triangles and use ...
3
votes
2answers
157 views

Integral inequality $\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$

How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
3
votes
2answers
169 views

How prove this $\int_{a}^{b}xf(x)dx\le\int_{a}^{b}xg(x)dx$

let $f(x),g(x)$ is continuous on $[a,b]$,and such $$\int_{a}^{x}f(t)dt\ge\int_{a}^{x}g(t)dt,x\in[a,b)$$ and $$\int_{a}^{b}f(t)dt=\int_{a}^{b}g(t)dt$$ show that: ...
3
votes
2answers
239 views

integration inequality [duplicate]

Possible Duplicate: Proving Integral Inequality Suppose $f(x)$ is differentiable on $[0,1]$ , $f(0)=0$ and $1\geq f'(x) >0 $ Prove that $\displaystyle\left(\int_{0}^{1} ...
3
votes
2answers
63 views

Proof about boundedness of $\rm Si$

$\def\Si{{\rm Si}}$ I want to prove the boundedness of $$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$ as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to ...