# 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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### An integral inequality involving the Bernoulli polynomials

The classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{z\operatorname{e}^{t z}}{\operatorname{e}^z-1}=\sum_{j=0}^{\infty}B_j(t)\frac{z^j}{j!}, \quad |z|<2\pi. \end{...
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### Inequality with distribution and integral

Let $a \in (0,1)$ be the (unique) solution of: $\displaystyle \int_0^1 (\theta - a)e^{\dfrac{(\theta - a)^2}{\beta}}g(\theta)d\theta = 0$ (1), where $g(\theta)$ is a continuously differentiable ...
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### Maximal mean value of a differentiable function with small variation

Let $B$ be a real valued function on $[0, 1]$ such that $\|B'\|_{L_2}\le 1$ and $B(0) = 0$. Denote \begin{gather}\label{small var condition} \varepsilon = \int_0^1 B^2(s)\, ds - \left(\int_0^1 B(s)...
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### Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$ [duplicate]

I want to prove the inequality with analysis methods instead of method of PDE Actually we can consider the following PDE: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ ...
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### Some integral inequality question with convexity

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a convex and integrable function such that $\lim_{x\to \infty} f(x) - x$ exists and is finite. I am also given the fact that $\int_{0}^1f(x)dx=\frac{1}{2}$. ...
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### Limit representation of the Gamma function

I was going through the proof for the limit representation of the Gamma function where to prove the interchange of the limit and integral is justified, the author uses the following two relationships ...
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### $L^{p}$ norm of $\frac{|\nabla f|^{2}}{f}$

I have asked this question in mathoverflow https://mathoverflow.net/questions/461885/lp-estimate-for-frac-nabla-f2f . I think I should also ask here. I’m trying to obtain an $L^{p}$ estimate under ...
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