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

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29
votes
2answers
715 views

On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$ It's easy to ...
25
votes
1answer
476 views

Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

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: ...
20
votes
1answer
696 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 ...
15
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 ...
14
votes
2answers
220 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 ...
14
votes
2answers
570 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 ...
14
votes
1answer
318 views

Inequality of numerical integration $\int _0^\infty x^{-x}\,dx$.

There was a friend asking me how to prove $$\int_0^\infty x^{-x}\,dx<2$$ Mathematica showed that its approximate value is 1.99546, so I think it isn't easy to solve it, can you provide me some ...
12
votes
6answers
470 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, ...
12
votes
1answer
324 views

Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$ \int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}. $$ Below is my ...
11
votes
4answers
4k 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 ...
11
votes
4answers
209 views

Prove $1.43 < \displaystyle \int_0^1 e^{x^2} \mathrm{d}x < \frac{e+1}{2}$

Prove $$1.43<\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2}$$ What I did; As I have no idea how to approach the left inequality I work with $$\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2} \iff ...
11
votes
2answers
441 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
6answers
826 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 ...
10
votes
3answers
557 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 ...
10
votes
2answers
649 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 ...
10
votes
3answers
600 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 ...
10
votes
2answers
705 views

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

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
10
votes
2answers
611 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 ...
10
votes
1answer
223 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 ...
10
votes
2answers
242 views

Easier ways to prove $\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$

Prove that $$\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$$ One way to do this is use the idea in the proof of Sophomore's dream. We have $$x^{-x}=\exp(-x\log x)=\sum_{n=0}^\infty\frac{(-1)^nx^n\log^n ...
9
votes
8answers
191 views

Prove that $\int_a^bxf(x)dx\geq\frac{b+a}{2}\int_a^bf(x)dx$

Let $f:[a,b]\to\mathbb{R}$ be continuous and increasing, show that $$\int_a^bxf(x)dx\geq\frac{b+a}{2}\int_a^bf(x)dx$$ I am thinking of using integration by parts. First let $$F(x)=\int_a^xf(t)dt$$ ...
9
votes
1answer
115 views

How prove this inequality $\left|\int_{a}^{b}f(x)dx\right|\leq \frac{1}{8}(b-a)^{2}\int_{a}^{b}\left|f''(x)\right|dx$

Let $f(x)\in \mathcal C^2([a,b]),f(\frac{a+b}{2})=0$. Show that$$\left|\int_{a}^{b}f(x)dx\right|\leq \frac{1}{8}(b-a)^{2}\int_{a}^{b}\left|f''(x)\right|dx.$$ I try to use Taylor's Theorem with ...
9
votes
1answer
265 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 $\displaystyle f(1)=\int_0^1f(x)dx=0$. Prove that $$\left|\int_0^x f(t)dt\right|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|\,\,\,\,\,\, \forall x\in [0,1].$$ ...
9
votes
1answer
379 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 ...
8
votes
1answer
342 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 $$ ...
8
votes
3answers
391 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
2answers
141 views

Inequality of integrals $\int_0^1(f(x))^2 dx \geq 4$ if $\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$

If $$\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$$ prove that $$\int_0^1(f(x))^2 dx \geq 4$$ EDIT My attempt is as follows - I can use only the $\int xf(x)$dx part to get a bound $\int f^2(x) dx \geq ...
8
votes
3answers
242 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
225 views

How to prove that $\int_{-\pi}^{+\pi}\cos{(2x)}\cos{(3x)}\cos{(4x)}\cdots\cos{(2005x)}dx$ is positive

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
438 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
2answers
155 views

Two inequalities for $\left(\int_{a}^{b}|f(x)|^rdx\right)^{\frac{1}{r}}$

Show that if $f\in \mathcal C^{n+1}([a,b])$ and $f(a)=f^{'}(a)=\cdots=f^\left(n\right)(a)=0,$ then the following statements are ture: $\mathbf a)$ $ \forall r\in[1,\infty),$the inequality ...
8
votes
1answer
161 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 $$\int_0^1f^2(x)\,dx\ge\frac{2m-1}{m^2+6m-3}$$
7
votes
2answers
193 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
137 views

hard integral inequality with $\pi$

a) Prove that $\int_{0}^{\infty} \arcsin{\pi^{-x^2}} dx$ converges b) Prove that $\int_{0}^{\infty} \arcsin{\pi^{-x^2}} dx<1$ I have tried to find suitable integral sum for b), unsuccessfully. ...
7
votes
1answer
1k 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 ...
7
votes
1answer
130 views

Prove an integral inequality $|\int\limits_0^1f(x)dx|\leq\frac{1-a+b}{4}M$

Let $f$ be a differentiable function on $[0,1]$ and $a,b\in(0,1)$ such that $a<b$, $\int\limits_0^af(x)dx=\int\limits_b^1f(x)dx=0$. Show that: $$\left|\int_0^{1} ...
7
votes
2answers
76 views

Prove a function is in $L^2[0,1]$

If $f\in L^2[0,1]$, and $$g(x)=\int_0^1\frac{f(t)\mathrm dt}{|x-t|^{1/2}},\quad x\in[0,1],$$ show that $\|g\|_2\le2\sqrt2\|f\|_2$. I tried Minkowski's integral inequality (with $p=1/2$, so ...
7
votes
1answer
77 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
2k views

A function of two cumulative probability distributions with same first 2 moments

Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment (hence ...
7
votes
1answer
84 views

Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove, as an auxiliary step, the following deceptively simple inequality: $$E(X^a) E(X^{a+1} \ln X) > E(X^{a+1})E(X^a \ln X) $$ where ...
7
votes
0answers
134 views

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 ...
7
votes
1answer
497 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 ...
6
votes
2answers
180 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
336 views

How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$

Let $f\in C^{1}[0,1]$ such that $f(0)=f(1)=0$. Show that $$\left(\int_{0}^{1}f(x)dx\right)^2\le\dfrac{1}{12}\int_{0}^{1}|f'(x)|^2dx.$$ I think we must use Cauchy-Schwarz inequality ...
6
votes
3answers
132 views

How to prove the inequalities $\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}(\sin x)^ndx\ge 0$

Show that: $$\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}(\sin x)^ndx\ge 0$$ My idea:maybe $\sin{(x^n)}\ge (\sin{x})^n?$
6
votes
2answers
455 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
3answers
180 views

Show that $\left(\int_{0}^{1}\sqrt{f(x)^2+g(x)^2}\ dx\right)^2 \geq \left(\int_{0}^{1} f(x)\ dx\right)^2 + \left(\int_{0}^{1} g(x)\ dx\right)^2$ [closed]

Show that $$ \left( \int_{0}^{1} \sqrt{f(x)^2+g(x)^2}\ \text{d}x \right)^2 \geq \left( \int_{0}^{1} f(x)\ \text{d}x\right)^2 + \left( \int_{0}^{1} g(x)\ \text{d}x \right)^2 $$ where $f$ and $g$ ...
6
votes
2answers
252 views

A false integral inequality

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) d x = f\left(\frac{a+b}{2}\right) = 0$$ ...
6
votes
4answers
663 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 ...
6
votes
3answers
95 views

Bound on $f(0)^2$ by integrals of $f^2$ and $(f')^2$ on $[0,1]$.

Let $f$ be a function which is $C^1((0,1))\cap C([0,1])$. I would like to be able to show $$ \frac{1}{2}f(0)^2 \leq \int_0^1 f(x)^2dx + \int_0^1f'(x)^2dx $$ where we are assuming that $f$ is a ...