# Tagged Questions

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### 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 $f>0$.
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### inequality of integrals

Let $f$ be of bounded variation on $[0,1]$ and $g:[0,1]\rightarrow \mathbb{R}$ be Lebesgue integrable on $[0,1]$. Prove |\int_0^1fg d\lambda|\leq (|f(0)|+\text{Var}_{[0,1]}f)\cdot ...
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### Upsetting Integral inequality

I have two smooth scalar non-negative functions f,g in $\mathbb{R}^+$ such that $$f(x) \leq g(x), ~\forall x\in \mathbb{R}^+$$ which are integrable with finite integrals in $\mathbb{R}^+$. I would ...
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### 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: ...
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### 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 ...
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### inequality with sup and Expectations

let $X_{t}$ Itô diffusion how we can get : $$\sup_{x\in \mathbb{R}^{n}}|E^{x}[f(X_{t})]| \leq \sup_{y \in \mathbb{R}^{n}}|f(y)|\sup_{x\in\mathbb{R}}E^{x}[1]=\sup_{y\in\mathbb{R}^{n}}|f(y)|.$$ Here ...
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### Prove that $\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$

Let $f$ be a continuously differentiable function on $[0,1]$ and $f(0)=0$. Prove that $$\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$$ Thank you!
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### 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 ...
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### prove this $\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$

let $f\in C^1[0,2]$,and such $\int_{0}^{2}f(x)dx=0,f(0)=f(2)$, show that $$\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$$ I think we must use $Cauchy$ inequality my idea:I have see this let ...
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### 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 ...
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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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### 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 ...
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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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### The integral $\int_0^1\dfrac{(-x)^n}{1+x} dx$

How can I prove that: $\forall x \in \mathbb{N}\setminus {0} \quad \dfrac{-1}{n+1}\le \int_0^1\dfrac{(-x)^n}{1+x} dx \le \dfrac{1}{1+n}$ $\lim_{n\to+\infty}\Sigma_{i=1}^{n}\dfrac{(-1)^{i-1}}{i}$. ...
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### How prove $\int_{0}^{1}F(x)dx\le \int_{0}^{a}G(x)dx$

if $G$ and $F$ are integrable,$a>0,G(x)\ge F(x)\ge 0$,and $$\int_{0}^{1}xF(x)dx=\int_{0}^{a}xG(x)dx$$ show that $$\int_{0}^{1}F(x)dx\le \int_{0}^{a}G(x)dx$$ ...
### Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$
Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$ My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...