1
vote
2answers
45 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
2
votes
0answers
54 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
0
votes
2answers
39 views

Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? \begin{equation} f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...
1
vote
0answers
49 views

L1 norm of a trigonometric polynomial

For a real $x$, $f(x) = \sum_{k=-T}^{T}e^{ikx}$ is the well known Dirichlet kernel. It is also known that $\|f\|_{L_1}=\int|f(x)|dx \le C_1\log T + C_2$ for some $C_1,C_2$ independent of $T$. ...
0
votes
1answer
22 views

An integral inequality question.

If we have two functions $f,g:[a,b]\to\mathbb{R}$ and we know they are bounded, so: $\sup_{x\in[a,b]}|f(x)|=K$, and $\sup_{x\in[a,b]}|g(x)|=M$. Where $K,M$, are positive finite constants, which of ...
3
votes
2answers
70 views

$\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$ $\Rightarrow$ $\left|f(x)\right|\le \sqrt3$

Let $f:[0, 1]\rightarrow \mathbb{R}$ be a function that is continous on $[0,1]$ and derivable on $(0, 1)$. If $\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$, show that $\left|f(x)\right|\le ...
1
vote
0answers
55 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 ...
0
votes
1answer
35 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
2
votes
2answers
81 views

Trigonometric Inequality $\cos 1 +\cos2+\ldots +\cos n < 0.55$ can be solved with the help of Integrals?

How can I prove for every $n \in \mathbb{N}$ $$\cos 1 +\cos2+\ldots +\cos n < 0.55$$ Any idea, any solution? Thanks! EDIT Can be solved this inequality with the help of integrals, because I met ...
0
votes
2answers
79 views

Show that if $\displaystyle\int_0^1f(x)dx=a$, then $\displaystyle\int_0^1\sqrt{f(x)}dx\ge a^{2/3}$

$f$ is continuous on $[0,1]$ and there is $a>0$ such that, $0\le f(x)\le a^{2/3}$ for $x\in[0,1]$. Show that if $\displaystyle\int_0^1f(x)dx=a$, then $\displaystyle\int_0^1\sqrt{f(x)}dx\ge ...
1
vote
1answer
21 views

Proof inequality for quadrature

I have a short question. how can i proof $\int_{a}^b \vert \alpha x + \beta \vert^2 \, dx \leq (b-a) \left( \vert \alpha a + \beta \vert^2 + \vert \alpha b + \beta \vert^2 \right)$.
3
votes
2answers
76 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 $$ ...
13
votes
1answer
243 views

A beautiful inequality for convex functions

Let $f\in \mathcal{C}([0,1],\mathbb R_+)$ increasing. Prove that there exist $g,h\in \mathcal{C}([0,1],\mathbb R)$, convexs, such that $g\leqslant f \leqslant h$ and : $$\displaystyle ...
3
votes
3answers
52 views

Is the following inequality true?

Suppose that $\int_{0}^{1}|f(x)|dx<\epsilon.$ Is the following inequality true $$ \frac{1}{|I|}\int_{I}|f(x)|dx\leq \epsilon $$ for any subinterval $I\subset [0,1].$
1
vote
1answer
29 views

help with integral inequality

Let $P(R)=e^R\cdot\int_R^{\infty}F(z)e^{-z}dz=\int_0^{\infty}F(R+z)e^{-z}dz$. Is it true that $P(R) \geq 0$ for all $R$ implies $F(z) \geq 0$ for all $z$? In my case, $F(z)$ is a difference of CDF ...
2
votes
0answers
63 views

Name of theorem?

I am trying to understand a proof which uses the following statement without further explanation, so I am wondering if this is a well known theorem? For the unit ball $B$ with radius $r>0$ and the ...
0
votes
1answer
47 views

Prove that $x \rightarrow \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$ is convex

To put it bluntly I'm stuck proving proving the subsequent inequality $$ \forall x>0, \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt \int_0^\infty \frac{t^2 e^{-tx}}{1+e^{-t}}dt \geq {\left ( ...
2
votes
1answer
67 views

Cauchy-Schwarz-like inequality of integrals

Let $f,g,$ be integrable on $[a,b]$. Prove that $$\int_a^b(fg)^2\le\int_a^bf^2\int_a^bg^2$$ I know that from Cauchy-Schwarz we have $$\left(\int_a^bfg\right)^2\le\int_a^bf^2\int_a^bg^2$$ so if we ...
2
votes
1answer
27 views

Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
0
votes
1answer
30 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
7
votes
1answer
114 views

Compare the integrals $\int_0^{\frac{\pi}{2}}\sin(\cos x)dx$ and $\int_0^{\frac{\pi}{2}}\cos(\sin x)dx$

Compare the following two integrals: $$\int_0^{\frac{\pi}{2}}\sin(\cos x)dx,\quad \int_0^{\frac{\pi}{2}}\cos(\sin x)dx$$ First I observe that by making the change of variable $x=\frac{\pi}{2}-x$,we ...
1
vote
3answers
92 views

Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite?

Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite? So I know that $| \int_0^1 g(x)dx | \leq \int_0^1 |g(x)|dx = ...
3
votes
2answers
126 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 $f>0$.
1
vote
1answer
48 views

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? ...
4
votes
4answers
98 views

Inequality $\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$

Let $f$ be a positive continuous function defined on a closed interval $[0,1]$, then it is true that: $$\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$$ I tried to show ...
7
votes
1answer
135 views

Understanding why $\int_0^{\pi/2} \sqrt{1+\cos^2x} \geq \frac{\pi}{4}\bigl( 1 + \sqrt{2}\bigr)$

Lately I stumbled accros the magnifient paper by Roger Nelsen, which can be found here Symmetry and Integration In this paper it is shown that $$ \int_0^{\pi/2} \frac{\mathrm{d}x}{1 + ...
0
votes
1answer
27 views

Area in the plane described by inequalities

An area in the plane is specified by the following inequalities: $$x^2 \le y \le \frac6{\sqrt{x}}, \; x\ge 1$$ How do I: draw this area? math this area? Any ideas?
0
votes
0answers
70 views

$\int_{-1/2}^{1/2}|\sum_{k=1}^{n}\cos (k\pi x)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi k 2^{-i}}|dx<1?$

Now we have an integral. Does the following inequality hold? $$\int_{-1/2}^{1/2}\Bigg|\sum_{k=1}^{n}\cos (k\pi x)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi k 2^{-i}}\Bigg|dx<1$$ If doesn't, ...
2
votes
0answers
43 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
1
vote
0answers
63 views

Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
1
vote
0answers
22 views

Inequality: double jensen, is this correct?

Imagine I have an exponent $a=bc$ where $b>1$ and $0<c<1$, the product $a=bc\in \mathbb{R}$ might be greater, equal or less than 1. Then for a measurable set with measure 1. Can I say the ...
1
vote
2answers
91 views

Absolute value bound of Lebesgue integral

For the Riemann integral, we have the bound $$\left|\int_Af(x)dx\right|\leq\left(\sup_{x\in A}|f(x)|\right)\cdot\left|\int_Adx\right|$$ Do we have a similar bound for the Lebesgue integral, one like ...
0
votes
1answer
60 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
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 ...
0
votes
0answers
75 views

Inequality (related to Hölder's inequality?)

Is there any way to put the outer exponent inside the integral when the set we integrate over doesn't have finite measure? i.e. $$\left(\int_{\mathbb{R}} f(x)dx\right)^{\alpha}\leq C_{\alpha} ...
0
votes
0answers
30 views

How to prove a duality of $L^p$ spaces? [duplicate]

Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $f:\Omega\longrightarrow \mathbb{R}$ be a measuable function. Let $1\leq p< \infty$ and $1/p+1/q=1$. Prove that the following are equivalent: ...
1
vote
0answers
67 views

Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$ \int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1, $$ where $J_2(x)$ is a Bessel ...
0
votes
1answer
74 views

Proof of $|\int _a ^b \mathbf f | \leq \int _a ^b |\mathbf f|$

Let $\mathbf f:[a,b]\rightarrow \mathbb R ^n$ be continuous. I'm trying to prove the following fact: $$|\int _a ^b \mathbf f| \leq \int _a ^b |\mathbf f|,$$ where: $$|\mathbf x|^2=\sum _{i=1} ^n x_i ...
1
vote
5answers
175 views

How prove this? $\dfrac{\int_{a}^{b}f(x)dx}{b-a} \ge \dfrac{\max f(x)}{2} , a<b$

Let $f(x)\in R$ be a concave function then show that $$\dfrac{\int_{a}^{b}f(x)dx}{b-a} \ge \dfrac{\max f(x)}{2} , x\in [a,b],a<b$$ I have $$M=max f(x)$$ $$\int_{a}^{b}f(x)dx \le \int_{a}^{b}M$$ ...
0
votes
0answers
34 views

Integral inequality (quotient difference)

I would like to prove the following: $$\int_{-1-x_2}^{1-x_2}\int_{-1-x_1}^{1-x_1}\frac{|f(x_1+h_1,x_2+h_2)-f(x_1,x_2)|^2}{|(h_1,h_2)|^{3}}dh_1dh_2\geq ...
1
vote
1answer
33 views

How can I show this function is bounded?

Let $p>1$ and define $F_p:(-1,1)\times (-1,1)\to\mathbb{R}$ by $$F_p(a,b)=\frac{1}{|a|}\int_{-1+b}^{1+b}\frac{1}{\sqrt{1+\frac{x^2}{a^2}}^{p+1}}dx$$ I would like to show that this function ...
2
votes
1answer
38 views

Comparison of semi-norms

Let $p\in (1,\infty)$ and define $$|f|_1=\left(\int_0^1\int_0^1\frac{|f(x)-f(y)|^p}{|x-y|^p}\,dx\,dy\right)^{1/p}$$ ...
2
votes
2answers
74 views

Help with a simple problem involving a functional inequality (trying to prove Gronwall's inequality)

So while trying to prove Grownwall's inequality, my proof led me to the following statement: $h'(x) \le h(x)g(x)$. Now when $h'(x)=h(x)g(x)$ the following holds: $h(x)=k \exp G(x)$, where $k$ is a ...
2
votes
1answer
146 views

Obtain lower and upper bounds

How do I obtain upper and lower bounds for a summation function: $$ \sum_{i=1}^{25}i^4 $$ Somehow it involves an integral: $$ \int_{0}^{25}x^4\:\mathrm{d}x $$ If you solve it it gives $1953125$ (this ...
0
votes
0answers
24 views

Sufficient condition for inequality related to mean, median, or other standard object?

I have an mathematical model of citations to journal articles. The model looks at what happens when an article is put on a platform (like arXiv) that allows better cross referencing. It shows that ...
1
vote
0answers
46 views

convolution inequality on R

Let $\nu$ be a complex Radon measure on $\mathbb{R}$ such that $$ \int_{\mathbb{R}} \check{\overline{f}}*f\ d\nu\geq 0 $$ for any complex continuous function $f$ with compact support, where ...
1
vote
3answers
67 views

inequality of integral ratio

$a(x),b(x),c(x),$ and $d(x)$ are positive function of $x$. $\frac{a(x)}{b(x)}$ and $\frac{c(x)}{d(x)}$ increases in $x$. Moreover, we have $\frac{a(x)}{b(x)}<\frac{c(x)}{d(x)}$ holds for all x $\in ...
2
votes
2answers
101 views

Generalization of Jensen's inequality for integrals?

Jensen's inequality for sums says that for $f$ convex, $$f\left(\sum_1^n \alpha_i x_i\right)\leq \sum_1^n \alpha_i f(x_i), \,\,\,\,\text{for } \sum_1^n \alpha_i = 1.$$ I have read that a ...
1
vote
2answers
122 views

An integral inequality.

I want to know that is it possible to show that $$ \int_{0}^{T}\Bigr(a(t )\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}} $$ for some $C>0$ where $a(t)>0$ and ...
5
votes
0answers
148 views

Evaluate integral $ I_s(x) \leq \frac{C}{(\pi(1-2 \alpha s))^{d/2}}\exp\left(\frac{\alpha}{1-2 \alpha s }|x|^2\right) $

For all $ x \in \mathbb{R}^n ,\hspace{5mm} 0 \leq s<t ,\hspace{5mm} t \in \mathbb{R}^+$ $$ I_s(x)=\int_{\mathbb{R}^n}\left|v\left(y \sqrt{2s}+x\right)\right|\exp(-|y|^2) \, \mathrm dy. $$ How we ...