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

learn more… | top users | synonyms

17
votes
1answer
645 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
274 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
518 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 ...
11
votes
6answers
377 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, ...
9
votes
4answers
1k 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
470 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
542 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
2answers
531 views

How prove this $\frac{2}{\pi}\int_{x}^{px}\left(\frac{\sin{t}}{t}\right)^2\,\mathrm dt\le 1-\dfrac{1}{p}$ 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 ...
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
451 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
2answers
178 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
415 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
131 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
207 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
191 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 ...
7
votes
2answers
137 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
198 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
396 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
136 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
111 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
2answers
219 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 ...
5
votes
1answer
180 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: ...
5
votes
1answer
68 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
766 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
0answers
223 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 ...
5
votes
1answer
357 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
2answers
146 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 ...
4
votes
1answer
86 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
1answer
105 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: ...
4
votes
2answers
206 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
182 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
169 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
133 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
131 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
152 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
57 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 ...
3
votes
1answer
172 views

Prove that $\int_0^1f(x)dx$$\int_0^1g(x)dx\ge1$.

Suppose $f(x)$ and $g(x)$ are positive measurable functions defined on $(0,1)$, satisfying $f(x)g(x)\ge1$ for any $x\in(0,1)$. Prove that $\int_0^1f(x)dx$$\int_0^1g(x)dx\ge1$. Totally no idea about ...
3
votes
1answer
120 views

Cauchy inequality $\Rightarrow$ Schwarz's integral inequality. Why can't the limit of the prior be used to deduce the latter?

Given the Cauchy-Schwarz inequality and the Riemann definition for the integral, $$\sum_{k=1}^{n}a_kb_k\le\sqrt{\sum_{k=1}^{n}a_k^2}\sqrt{\sum_{k=1}^{n}b_k^2}$$ $$\int_a^bf(x)dx=\lim_{n\rightarrow ...
3
votes
1answer
168 views

Hilbert's Inequality

Could you help me to show the following: The operator $$ T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy $$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p $$ for $1 <p< \infty$ where ...
3
votes
2answers
127 views

An integral inequality related to Taylor expansion

Problem. Let $f:[a,b]\to\mathbb{R}$ be a function such that $ f\in C^3([a,b])$ and $f(a)=f(b)$. Prove that $$ ...
3
votes
2answers
272 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 ...
3
votes
1answer
139 views

How prove this inequaliy $\dfrac{1}{b-a}\int_{a}^{b}|f(x)|dx>\dfrac{1}{2}|f(a)+f(b)|$

let $f(x)$ have twice differentiable on $[a,b]$,and such $$f(x)\cdot f''(x)<0$$ show that $$\dfrac{1}{b-a}\int_{a}^{b}|f(x)|dx>\dfrac{1}{2}|f(a)+f(b)|$$ I only know and can prove follow this ...
3
votes
1answer
219 views

integral inequality with derivative

Let a function $f:[0,1]→\mathbb{R}$ have a continuous derivative and $$\int_{0}^{1}f(x)dx=0$$ Show that for every $\alpha \in [0,1]$, $$\left|\int_{0}^{\alpha}f(x)dx\right|\le\frac{1}{8} \mathbb{sup} ...
3
votes
2answers
79 views

Proof of Wirtinger inequality

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ ...
3
votes
1answer
218 views

How to prove $|xe^{-x^2}\int_{0}^{x}e^{t^2}dt-ye^{-y^2}\int_{0}^{y}e^{t^2}dt|<|x-y|$ for $x\ne y$

Let $x,y$ be distinct positive numbers. Show that $$\left|xe^{-x^2}\int_{0}^{x}e^{t^2}dt-ye^{-y^2}\int_{0}^{y}e^{t^2}dt\right|<|x-y|.$$ I found this problem when I was dealing with the ...
3
votes
2answers
76 views

Find min $P$: $P=\frac{1}{(a+b)(b+c)}+\frac{1}{(c+a)(a+b)}+(c+1)(3+a+b)$

Let $a,b,c\geq 0$ and $a+b+c=1$. Know that never have two numbers both zero. Find min $P$: $$P=\frac{1}{(a+b)(b+c)}+\frac{1}{(c+a)(a+b)}+(c+1)(3+a+b)$$
3
votes
1answer
124 views

Integral inequality

Let $f$ be a continuously differentiable real-valued function on $[0,b]$, where $b>0$, with $f(0)=0$. Prove that $$\int\limits_0^b\frac{f(x)^2}{x^2}dx\leq4\int\limits_0^b f'(x)^2dx.$$ Thank you!
3
votes
1answer
186 views

A variation of Jensens inequality .

The following version of Jensens inequality is used in lecture notes, But i don't seem to get it . if $\phi : \mathbb R^n \to R$ is a convex function and $f_i \in L^1 (\Omega) $ for all $i$ and ...
2
votes
2answers
212 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} ...