# 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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### Integral Inequality involving a binomial

Let $\binom{t}{k}$ where $k\in \mathbb{N}$ be defined as $\frac{t(t-1)\cdots (t-k+1)}{k!}$. Prove that $\int_n^{\infty} \binom{t-1}{n-1} e^{-t} dt \le \frac{1}{(e-1)^n}$ My progress is as follows: we ...
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### Show that $\int_2^{+ \infty} \frac{\log(t)^2}{t(t-1)}dt \leq 4$

I couldn't prove this inequality $$\int_2^{+ \infty} \frac{\log(t)^2}{t(t-1)}dt \leq 4$$ I've tried integration by parts but it doesn't work.
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### What is the correct version of the Gronwall lemma? Can the sign of u(t) be variable?

In https://encyclopediaofmath.org/wiki/Gronwall_lemma the various forms of the Grönwall lemma in integral form are stated for NON NEGATIVE function $\phi$ And this coincides with what is written my ...
1 vote
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### Integration over the intersection of a hypersphere and half spaces

For $x\in\mathbb{R}^n$, $u\in\mathbb{R}^n$ and $0<c<\Vert u\Vert$, how could I compute (or find the upper bound of) the following integral $$\int_{x^\top x =1,~\vert u^\top x\vert\leq c}1dx.$$ ...
1 vote
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### Derivation of Hölder Inequality through Young's Inequaliy

I am having trouble following a proof where Young's Inequality is being used to derive Hölder's Inequality. More precisely, there is a particular and final step that utilizes integration in order to ...
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### Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$

The number of continuous functions $f:\left[0,\frac{3}{2}\right]\rightarrow (0,\infty)$ satisfying the equation$$4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$$ ...
1 vote
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### Prove that $x/((e+x)\ln(e+x)) \leq \ln(\ln(e+x)) \leq x/e$ for all $x > 0$

I've been working on a problem but have been stuck for several hours finishing it. The problem is to show that $$\frac{x}{(e+x) \ln(e+x)} \leq \ln(\ln(e+x)) \leq \frac{x}{e}$$ for all $x > 0$. ...
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### A fractional Gronwall lemma

I have an energy estimate of the form $$\dot{u} + \lambda u^{1+ \delta} \le C (1+t)^{-\sigma},$$ where $u=u(t)$ is positive, $\lambda, \delta > 0$ and one can assume $\sigma > 1$ large. I expect ...
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### An apparently simple but somehow unexpected inequality between integrals

Let $f:[0,1]\to[0,1]$ be any Lebesgue integrable function. Then $$I_1= \left(\int_0^1f(x)\sqrt x dx\right)^2 \leq \frac{4}{3}\int_0^1f(x)x^2dx=I_2\,.\tag{\star}$$ At first sight, I would have ...
### $L^2$ norm and $L^{\infty}$ inequality for periodic smooth functions
Let $\varphi \in C^{\infty}(\mathbb{T}^n,\mathbb{C})$ (i.e. just smooth periodic complex-valued function) and $f \in C^{\infty}(\mathbb{T}^n,\mathbb{C}^m)$. Then I was wondering if the following is ... 