# 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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### Probs. 10 (a), (b), and (c), Chap. 6, in Baby Rudin: Holder's Inequality for Integrals

Here is Prob. 10, Chap. 6, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $p$ and $q$ be positive real numbers such that $$\frac{1}{p} + \frac{1}{q} =1.$$ ...
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### An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
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### Minimizing the value of integral under certain conditions

I have been working on the following minimization problem: Minimize $I(f) := \int_{-\infty}^{\infty}{|f(x)|^2+|f'(x)|^2dx},$ where $f\in\mathcal{A} = \{f:\mathbb{R}\to\mathbb{R}$ is differentiable ...
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### Proof-Verification: $\int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x)\,{\rm d}x \leq \frac{1}{2019}\int_a^b f(x)\,{\rm d}x$

Problem Let $f(x)$ be continuous and increasing over $[a,b]$. Prove $$\displaystyle \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x){\rm d}x \leq \frac{1}{2019}\int_a^b f(x){\rm d}x.$$ Proof By ...
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### $\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{s}{n^{s+1}}$ looking for different proof

Let $n \in \mathbb{N}_{>0}$ and $s \in \mathbb{R}_{>0}$ then I am interested in the following inequality : $$\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{s}{n^{s+1}}$$ Here ...
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### Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
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### How does this inequality imply this one?

I am having a little trouble understanding this part of a proof. There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$ Now, $\text{J}_0 = \frac{\pi ^3}{24}$ The part of ...
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### Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...
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### Question about the assumption of a version of Grönwall's inequality.

According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following: Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ ...
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### Equality regarding power series and an integral

Given the power series $f(z)=\sum^\infty_{n=0}c_n(z-z_0)^n$ I have to prove that $$\int_0^{2\pi}|f(z_0+re^{it})|^2\;dt =\sum^\infty_{n=0}|c_n|^2r^{2n}$$ Here $r>0$ is such that \$\overline{B_r(z_0)...