Questions tagged [integral-inequality]

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

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Looking for name/references for basic integral inequality

I am looking for a reference (or theorem name) for the fact that an integral is larger if its non-negative integrand and region of integration are both strictly larger. I am looking at this ...
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Inequality on expectation

I will show the exact inequality I'm dealing with. So, say that I have a collection of random variables on a probability space $\{X_i\}_i^N: X_i=V_{n,i}/\sqrt{d_n}\in L^2(P)$. I think that this ...
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Visual interpretation of integral inequality

Is there any nice visual proof of the following proposition? Let $$F(x) = (p(x), q(x))$$ then $$||\int_0^t F \|| \leq \int_0^t ||F||$$ were $$\int_0^t F \$$ is a vector that results from ...
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Question about an Integral inequality (with norm, Holder or Minkovski)

I'm reading an article that associated with Integral type inequality, suppose $$B\left(x\right)\in L^{1}\left(0,T;W_{\mathrm{loc}}^{1,\alpha}\left(\mathbb{R}^{N};\mathbb{R}^{N}\right)\right),$$ in ...
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Inequality involving integrals of trigonometric functions

Prove the inequality $$\left|\int_0 ^{\pi/4} \frac{\tan x~dx}{3-\sin(x^2)}\right|≤ \frac{1}{4}\log_e 2.$$ I have tried many different ways to get this inequality but failed.
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upper bounds for $\int_a^{b} \frac{\exp(x)}{x}\ dx$

Let $a<b$ be a positive real numbers. Are there tight upper bounds for $\int_a^{b} \frac{\exp(x)}{x}\ dx$, specially asymptotic bounds when $a, b,\frac{b}{a}\to\infty$?
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How to prove the following inequalities.

The first inequality is $\int_{0}^{\infty} x^af(x)dx \leq a(\int_{0}^{\infty} xf(x)dx)$ for $0<a<1$ and $\int_{0}^{\infty} x^af(x)dx \geq a(\int_{0}^{\infty} xf(x)dx)$ for $a>1$ The ...
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under which situation inequality between two integral yeilds pointwise inequality?

suppose I have $$\int{f(x)} < \int{g(x)}$$ when I can conclude this : $f(x_{0})< g(x_{0})$ for some $x_{0}$ or is there such a $x_{0}$? how I can find it? if there is not answer, how ...
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Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Let $f:[0,1]\to[1,3]$ be continuous. Prove $$1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}.$$ The left is just Cauchy's inequality with integral form, but ...
I have a question about the proof of the inequality. The well known result stats Let $Z$ be a RV and let $0<s<t$. Then $$E(|Z|^s)^{1/s} \leq E(|Z|^t)^{1/t}$$ The proof follows almost ...
Problem description Let $h(t) > 0$ be a continuous real function and $x(t) \in \mathbb{R}^{3}$ be also a continuous function. Let $$T(t) = h + \dot{x}^T\dot{x}$$ It is known that  \dot{T} = -\...