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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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 $\... 0answers 91 views Integral inequality related to derivation While trying to understand a proof, i have stumbled upon the following statement: Let$f \in L^p(a,b)$be a$p-integrable function. Then the inequality \liminf_{s \rightarrow t} \frac{1}{t-s} \... 0answers 405 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 30 views Conditions to ensure an integral to be positive Suppose that g:[0,1]\to\mathbb{R} is a differentiable function with g(0)=0. I want to find conditions on g such that \int_0^1 f(x)g(x)\int_x^1f(t)\,dt\,dx=\int_0^1f(t)\int_0^xf(x)g(x)\,dx\,... 0answers 50 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 25 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 $$F(r):= \... 0answers 18 views Linear version of Gronwall's inequality, proof I am reading a proof of the following theorem: Assume \phi is a continuous function in [0,T] that satisfies$$\phi(t) = \alpha + \int_0^t (\beta \phi(s) + \gamma)ds, \hspace{0.5mm} t\in [0,T], $$... 0answers 17 views Finding discrete solutions to inequality involving Exponential Integral I want to identify the least natural number n (of course, it suffices to solve this problem for the reals, and then take the floor) such that$$-c \text{Ei}\left(-e^{\frac{a-d}{c}} (n+1)\right)+a-... 0answers 77 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 72 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[-...
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)...