# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$
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### 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 $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$
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### 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 ...
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### 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?
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### Proving an estimate for this integral

How can I show that$$\sqrt[3]6>\int_1^\infty\frac{(1+x)^{1/3}}{x^2}\mathrm dx?$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### An Integral inequality

Let $f$ be a continuously differentiable function on $[0,1]$. Prove that $$|f(\frac{1}{2})|\leq\int\limits_0^1|f(x)|dx+\frac{1}{2}\int\limits_0^1|f'(x)|dx$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$I_{n}\leq \frac{1}{n-1}\left(1-\frac{1}{e^{n-1}}\right)?$$ ...
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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 $n \geq 2$, $$... 3answers 132 views ### Squeeze an integral Would you have any idea about this problem ? Prove that for all nonnegative integers n, the following inequalities hold:$$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
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 ...
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.