Tagged Questions
7
votes
1answer
140 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 ...
0
votes
0answers
72 views
Simplifying $\frac{1}{n}\sum_{k=1}^n f(\frac{1}{k})$
Suppose that $$\displaystyle \forall x\in \mathbb{R}_+^* \quad f(x)=\frac{x^2-1}{4}-\frac{\ln(x)}{2}.$$
How can I simplify this: $$I(n)=\frac{1}{n}\sum_{k=1}^n f\left(\frac{1}{k}\right)$$
and prove ...
9
votes
2answers
210 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
...
3
votes
1answer
42 views
How to show a basic integral inequality?
The following inequality is quite clear for $R^1$:
$$\int_{B_1}1/|x-y|^\alpha dx\leq\int_{B_1}1/|x|^\alpha dx,\quad\forall y\in B_1,$$
where $B_1$ is the unit ball in $R^1$, i.e., $[-1,1]$ and ...
0
votes
0answers
77 views
Tight Upper/Lower bound for Incomplete Gamma function
Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions:
$$
\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t
$$
or
$$
\gamma(s,x) ...
1
vote
0answers
25 views
Bounds for the exponential integral
In Abramowitz and Stegun: Handbook of Mathematical Functions
(on page 229, property 5.1.20) it is found that
$$
\frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
1
vote
2answers
151 views
Cauchy-Schwarz Inequality for Integrals for any two functions clarification
I'm trying to work through a homework set, and it states that for any two functions, $f$ and $g$, that the following inequality holds:
$$
\int{fg} \le ||f|| \cdot ||g|| \le \frac{c}{2}||f||^2 + ...
8
votes
3answers
193 views
Funny integral inequality
Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$
and how to find the smallest ...
2
votes
1answer
66 views
An integral inequality with respect to $f$ and $f'$
Assume $f \in C[0,1]$ and $f(0)=f(1)=-\frac{1}{6}$,show that:
$$\frac{1}{4}+2\int_0^1f(x)\text{d}x\leq \int_0^1(f'(x))^2\text{d}x$$
1
vote
2answers
90 views
Inequality with integral
Assume that F is a strictly increasing and continuous function , differentiable if needed, over [0,a] such that F(0)=0 and F(a)=1, a>0. Prove or disprove :
$\int_0^a (F(x)-4F^2(x)+3F^3(x))\, ...
3
votes
2answers
226 views
$3\int_{0}^{1}(f'(x))^2dx \geq (2\int_{0}^{1}f(x)dx)^2 \impliedby 2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$
Let $f : \mathbb{R} \to \mathbb{R} $ be a differentiable function. Suppose that $2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$
Show that $$3\int_{0}^{1}(f'(x))^2 ...
2
votes
2answers
127 views
How can I prove these inequalities?
Let $f:\mathbb{R}_+\rightarrow\mathbb{R}$ be a $C^1$ function such that $f(0)>0$ and there exist an increasing sequence of positive reals $a_1,...,a_k$ with $k\geq 2$ satisfying $f(a_k)=0$.
Define ...
3
votes
2answers
100 views
Given $\int_0^1 x f(x)dx=0$, show that $\int_0^1|1-f(x)|dx>1/2$
I have seen this statement before, and I would like to use it in a proof I am working on. I do not quite remember the condition on $f$--whether it is just integrable or continuous.
Can someone point ...
2
votes
1answer
171 views
Proving Hölder's Inequality
Let $f,g,\alpha:[a,b]\rightarrow \mathbb{R}$ with $\alpha$ increasing and $f,g \in \mathscr{R}(\alpha)$, and $p,q>0$ with $\frac{1}{p}+\frac{1}{q}=1$. Prove that $$\left|\int_a^b ...
2
votes
4answers
100 views
prove or disprove that $\int_a^b |f(x)| \mathrm{d}x\geq |\int_a^b f(x)\mathrm{d}x |$
+Let $f$ be a continuous and integrable function over $[a;b]$, Prove or disprove that :
$\displaystyle\int_a^b |f(x)|\ \mathrm{d}x\geq \left | \int_a^b f(x)\ \mathrm{d}x\right|
$
2
votes
1answer
100 views
A trigonometric integral inequality
$$\displaystyle\frac{4\sin 1}{\pi }<\int_{0}^{1}{\frac{\cos x}{\sqrt{1-{{x}^{2}}}}}\text{d}x\le \frac{\pi }{2}\ln \left( \sec 1+\tan 1 \right)$$
I've got no ideas for this one.
2
votes
1answer
71 views
Integral inequality with sin exp
For $\displaystyle f(x)=\int_x^{x+1}\sin (\text{e}^t)\text{d}t$
Prove that :
$\displaystyle \text{e}^x\left|f(x)\right|\le 2$
1
vote
0answers
64 views
Inequalities of integrals of periodic functions
I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
2
votes
2answers
104 views
integration inequality [duplicate]
Possible Duplicate:
Proving Integral Inequality
Suppose $f(x)$ is differentiable on $[0,1]$ , $f(0)=0$ and $1\geq f'(x) >0 $
Prove that $\displaystyle\left(\int_{0}^{1} ...
1
vote
1answer
74 views
About implicit differential equation?
$\displaystyle \begin{align*}
& 0<x<1\wedge f\left( x \right)=\int_x^1 \frac{\left( 1-t \right)^2}{t^2} \text{d}t \\
& \text{Prove}:\ \ f\left( x \right)\ge \frac{2\left( 1-x ...
2
votes
1answer
73 views
approximate error between integral an sum
I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by ...
0
votes
1answer
38 views
Showing that $ \int_b^u \frac{\mathrm dx}{\ln x}\leq \frac{2u}{\ln u},e^2<b<u$
How can I show that:
$$ \int_b^u \frac{\mathrm dx}{\ln x}\leq \frac{2u}{\ln u}$$
where:
$$ e^2<b<u$$
?
1
vote
0answers
72 views
second order stochastic dominance
Let the nonnegative random variables $X$ and $Y$ have distribution functions $F$ and $G$ and density functions $f$ and $g$, respectively.
Suppose $X$ is second-order stochastically dominant over $Y$, ...
0
votes
1answer
65 views
Upper bounds for an integral with an infinite upper limit
I'm trying to work out an upper bound for the following problem, but I'm making very little progress. Hopefully, someone will be able to make a suggestion.
The integral I'm attempting to bound is:
...
4
votes
1answer
58 views
Sign of integral with increasing power
Maybe a trivial question. I think it is true, but I do not succeed in proving.
Suppose $h(x)$ is such that for all $a\geq 0$
$$
\int_a^\infty h(x) dx\geq0
$$
Prove for all $a\geq 0$
$$
\int_a^\infty ...
7
votes
1answer
149 views
If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$
If $f$ is a positive, monotone decreasing function, prove that
$\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$
3
votes
1answer
121 views
Integral Inequality $|f''(x)/f(x)|$
Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that
$$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$
1
vote
1answer
199 views
Looking for an inequality related to the Cauchy-Schwarz inequality
From the Cauchy-Schwarz inequality, we can prove that
$$\lVert w(x)\rVert^2_{L^2_{[0,1]}}=\int_0^1 w(x)^2\, dx \leq \sqrt{\int_0^1 w(x) \,dx}\cdot \sqrt{\int_0^1 w(x)^3\, dx}.$$
Is it possible to ...
0
votes
5answers
407 views
Why can't $\int_0^1\sin(x^2) dx$ be equal to $2$?
Why can't $\int_0^1\sin(x^2) dx$ be equal to $2$?
What makes this true? Intuitively, it makes sense. But why?
7
votes
5answers
193 views
Proving an estimate for this integral
How can I show that$$\sqrt[3]6>\int_1^\infty\frac{(1+x)^{1/3}}{x^2}\mathrm dx?$$
6
votes
1answer
147 views
Integral-Summation inequality.
The following question was in an entrance exam:
Show that, if $n\gt0$, then: $$\int_{{\rm e}^{1/n}}^{\infty}{\frac{\ln{x}}{x^{n+1}}\:dx}=\frac{2}{n^2{\rm e}}$$ You are allowed to assume ...
6
votes
2answers
298 views
Prove that $ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$
Prove the following integral inequality:
$$ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$$
This is a high school problem. So far i did manage to prove that the integral is bigger than $1.462$ by using ...
6
votes
4answers
580 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 ...
7
votes
1answer
374 views
Prove that: $ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$
I plan to prove the following integral inequality:
$$ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$$
Since we have to deal with a convex function on this interval i thought of ...
7
votes
4answers
177 views
Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
I posted this incorrectly several hours ago and now I'm back! So this time it's correct. Im trying to show that for $n\geq 1$:
$$\frac{2}{n+\frac{1}{2}} \leq ...
1
vote
0answers
88 views
Help proving that $\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$
I am trying to prove that for $n\geq 1$:
$$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+\sin^2(\frac{\pi}{t}) - \frac{2}{t}\sin(\frac{\pi}{t})\cos(\frac{\pi}{t}) + ...
2
votes
1answer
96 views
Inequality between Integral expressions
I want to prove that there exists a costant $C_{t,l}$ depending only on $t\in \mathbb{R}^+$ and $l\in \mathbb{Z}^+$ such that, for any
$d\in \mathbb{R}^+$, the following inequality of "Heat ...
7
votes
4answers
235 views
Prove that: $\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$
Here is another interesting integral inequality :
$$\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$$
According to W|A the difference between RS and LS is extremely small, namely 0.00241056. I ...
5
votes
3answers
167 views
Prove that: $\frac1{20}\le \int_{1}^{\sqrt 2} \frac{\ln x}{\ln^2x+1} dx$
I'm interested in proving the following integral inequality:
$$\frac1{20}\le \int_{1}^{\sqrt 2} \frac{\ln x}{\ln^2x+1} dx$$
According to W|A the result of this integral isn't pretty nice, and ...
2
votes
1answer
111 views
Another inequality with definite integrals
Let be $ f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous, monotone function. Then, if $a>0$ i must prove that the following inequality holds:
$$\int_{-a}^{a}xf(f(x)) \geq0$$
I wonder if there ...
1
vote
1answer
133 views
How do I go from a summation form to integral form of an expression?
How is the following inequality valid?
$$\sum\limits_{j = 1}^{N/2} \sum\limits_{n = j}^{\infty} \frac{1}{n^2} < \sum_{j = 1}^{N/2} \int_{j - 1/2}^{\infty} \frac{dx}{x^2}$$.
I came across this ...
4
votes
2answers
117 views
An Integral inequality
Let $f$ be a continuously differentiable function on $[0,1]$. Prove that $$|f(\frac{1}{2})|\leq\int\limits_0^1|f(x)|dx+\frac{1}{2}\int\limits_0^1|f'(x)|dx$$
2
votes
0answers
48 views
lower bound of a definte integral [duplicate]
Possible Duplicate:
Prove $\int_0^1 \left| \frac{f^{''}(x)}{f(x)} \right| dx \geq4$
Suppose $f\in C^2[0, 1]$, $f(0) = f(1) = 0$. For any $x\in(0,1)$, $f(x)\neq 0$.
Please show ...
1
vote
2answers
386 views
Inequality involving norm of matrix integral
This question seems basic but I could not find an answer. I have seen the inequality
$$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$
where $x(t) \in \mathbb{R}^n$ is a ...
1
vote
3answers
98 views
finding bound for the integral
I am trying to get bound for the following integral
$$
\int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty
$$
In particular, the bound of the form $\frac{constant}{r}$.
Sorry, we can ...
4
votes
1answer
94 views
Inequality for the integral $\frac{\ln x}{x^n}$
Define the integral $I_{n}$ as follows for $n$ an integer greater than $1$:
$I_{n}:=\int_{1}^{e}\frac{\ln x}{x^n}dx$
Is it true that
$$I_{n}\leq \frac{1}{n-1}\left(1-\frac{1}{e^{n-1}}\right)?$$
...
2
votes
1answer
87 views
Inequality for integral
Define the following integral with $n$ an integer greater than $1$:
$$I_{n}=\int_{0}^{1}\frac{e^t}{(1+t)^n}dt.$$
Is it true that for all $n \geq 2$,
$$ ...
3
votes
3answers
132 views
Squeeze an integral
Would you have any idea about this problem ?
Prove that for all nonnegative integers $n$, the following inequalities hold:
$$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
2
votes
4answers
489 views
Prove integral inequality
Assume that a function $f$ is integrable on $[0, x]$ for every $x > 0$.
Prove that for any $x > 0$, $\displaystyle\left (\int_{0}^{x}fdx \right )^2\leq x\int_{0}^{x}f^2dx$.
I have no idea ...
1
vote
2answers
103 views
Lower bound of an integral
How might I find a constant $k>0$ s.t. $\int\limits_0^1|f(x)|dx\geq k\max\{|f(x)|:x\in[0,1]\}$ for all continuous $f$ defined on $[0,1]$? Thank you.
