# 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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### $\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{s}{n^{s+1}}$ 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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How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where $\... 0answers 90 views ### Integral inequality related to derivation While trying to understand a proof, i have stumbled upon the following statement: Let$f \in L^p(a,b)$be a$p-integrable function. Then the inequality \liminf_{s \rightarrow t} \frac{1}{t-s} \... 0answers 389 views ### Weak/Variational Gronwall type inequality I came across the following weak differential inequality while looking through F.Otto's paper on L^{1} contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - \int_{... 0answers 65 views ### 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.$$Proof By ... 0answers 35 views ### Gagliardo-Nirenberg inequality for fractional Sobolev spaces Wikipedia states two versions of the Gagliardo-Nirenberg inequality for nonfractional Sobolev spaces. I'm interested in generalizations to fractional (Slobodeckij) Sobolev spaces. Such a ... 0answers 23 views ### Exponential decay and integration I am confronted with the following problem: Let \mu be a probability measure on \mathbb{R}. We wish to show that for any p \in \mathbb{N} and r \in \mathbb{R}, the integral$$ F(r):= \... 0answers 15 views ### Linear version of Gronwall's inequality, proof I am reading a proof of the following theorem: Assume\phi$is a continuous function in$[0,T]$that satisfies $$\phi(t) = \alpha + \int_0^t (\beta \phi(s) + \gamma)ds, \hspace{0.5mm} t\in [0,T],$$ ... 0answers 16 views ### Finding discrete solutions to inequality involving Exponential Integral I want to identify the least natural number$n$(of course, it suffices to solve this problem for the reals, and then take the floor) such that $$-c \text{Ei}\left(-e^{\frac{a-d}{c}} (n+1)\right)+a-... 0answers 56 views ### Why is the laplacian a closed operator in W^{2,p}(\mathbb{R}^n)? I have read that the laplacian is a closed operator in W^{2,p}(\Omega),(that is, \Delta : W^{2,p} \to L^p) where \Omega satisfies some conditions (I need the case \Omega = \mathbb{R}^n so ... 0answers 69 views ### Reference for Garding's Inequality Does anyone know which is the original paper in which Garding's inequality first appeared? By this, I mean https://en.wikipedia.org/wiki/G%C3%A5rding%27s_inequality. Many thanks! 0answers 28 views ### Calculating upper bound for increments of a function Let \varphi_{f,\delta} be a function defined by \varphi_{f,\delta}(u)=\sup\left\{\left|f(y)-f(x)\right|:x,y\in\left[u-\delta,u+\delta\right]\right\} for a bounded and symmetric function f:\left[-... 0answers 20 views ### Equality regarding power series and an integral Given the power series f(z)=\sum^\infty_{n=0}c_n(z-z_0)^n I have to prove that$$ \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)...
Let $f(x)$ is a continuous function in $[1;3]$ such that $\max\limits_{x\in[1;3]} f(x)=2$ and $\min\limits_{x\in[1;3]} f(x)=\dfrac{1}{2}$. Find the maximum value of \$\displaystyle P=\int\limits_1^3 f(...