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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inequalities AM-GM
Give $a,b,c\ge 0$ satisfy $a^2+b^2+c^2=3$. Prove that:
$$3(a+b+c+abc) \ge 4\sqrt{(a^2+bc)(b^2+ca)(c^2+ab)}.$$
I think use PQR (a+b+c=p, ab+bc+ca=q, abc=r). If I square LHS and RHS I got: $$27+18q+9r^2+...
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
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How to prove this integral inequality $\int_{ - 1}^1 {{{\left| {g\left( s \right)} \right|}^2}ds}$
Question
Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\...
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Example that the Poincare inequality fails for higher exponent
I am trying to bound a function $u$ on a ball $B\subset \mathbb R^n$ using the norm of its gradient.
I found in many literatures that $\lVert u-\bar u\rVert_{L^q(B)}$$\lesssim_ {p,q,n}\lVert\nabla u\...
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How to estimate $\int_0^{L/2} f(x)dx$ from $f(x)\ge 2x-\frac{L}{D} \int_0^x f(t)dt$
$D>0, L>0$ are constant. $f:[0, +\infty) \rightarrow \mathbb R$ is non-negative continuous function. $f(0)=0$ and $D=\int_0^{+\infty} f(x) dx$. In fact, when $x>0$ is large enough (than $L$),...
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Prove that $\left(\int_0^1 f(x)dx\right)^2\leq\frac{1}{12}\int_0^1\left(f'(x)\right)^2dx$ [duplicate]
The following problem is from Andreescu's "Problems in Real Analysis: Advanced Calculus on the real axis". More specifically it's problem $10.6.11$:
Let $f:[0,1]\rightarrow\mathbb{R}$ be a ...
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Integral inequality from AMM 1992
I would like to know the solution of the following 1992 AMM problem:
Let $f$ be a continuous non-negative function defined on the square $[0,1]^2$. Show that
$$
\int_0^1\int_0^1\int_0^1\int_0^1f(x_1,...
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Comparing two different weighted averages of a function. [duplicate]
I need to prove the following:
Assumptions:
$f(\cdot)$ and $g(\cdot)$ are continuous and weakly increasing on $[0,1]$.
$f(x)\geq 0$ and $g(x)\geq 0$ for $\forall x \in [0,1]$.
$\int_0^1g(x)dx = 1$.
...
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estimating expression
Let $\gamma(t) = e^{it}$ where $0 \le t \le 2\pi$.
Prove that
$$
\left|\int_{\gamma} \frac{1}{12+5z} dz \right| \le \frac{20}{91}\pi.
$$
I tried ML inequality and got this integral $\le \frac{2\pi}{7}$...
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Inverse inequality
I would like to prove the following inequality:
Let $f$ be a function $f:\mathbb R\to\mathbb R$ such that $0\leq f\leq 1$ on $\mathbb R$ and $f=1$ on $[1,+\infty)$ and $f=0$ on $(-\infty,-1]$. Prove ...
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A strict triangle Inequality $\left| \int_a^b f(t) dt\right| < \int_a^b |f(t)|dt$ for Integrals [duplicate]
The triangle inequality for integrals
$$\left| \int_a^b f(t) dt\right| \le \int_a^b |f(t)|dt$$
is well-known, where $f(t):\mathbb R\to \mathbb C$. However, I was wondering if there is a condition for ...
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Computable counterexample of an integral inequality
Statement 2: For every continuous $f : [0,\frac{\pi}{2}] \to (0,+\infty)$ satisfying $\int_0^{\pi/2} f(t)\sin(t)dt < f(\frac{\pi}{2})$, then
$$
\int_0^{\pi/2} \sqrt{f(t)}dt < \frac{\pi}{2}\sqrt{...
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Relation of $L^1$ and "$L^{1/2}$"
For any continuous $g:[0,\frac{\pi}{2}]\rightarrow (0,+\infty)$ and $\forall \varepsilon >0$, is there a contiuous $h(t):[0,\frac{\pi}{2}]\rightarrow (0,+\infty)$ such that
$$
h(\frac{\pi}{2})\le\...
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Commutator estimate (PDE)
While studying uniqueness of a weak solution to KdV equation (on torus), I encountered a problem to bound
$$
\int_{\mathbb{T}}f(x)^{2} \partial_{xxxxx}(\mathbb{P}_{N}f)(x) dx
$$
using $\|f\|_{H^{2}(\...
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$L^1$ norm and an integral inequality
Assume $f : [0,\frac{\pi}{2}] \to (0,+\infty)$ is continuous, and for $\epsilon>0$ small enough, there is
$$
\left(\frac{1}{\pi/2}\int_0^{\pi/2} \sqrt{f(t)}dt\right)^2 > \epsilon+ \int_0^{\pi/2} ...
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$\int_0^{\pi/2} f(t)\sin t dt =\int_{\pi/2}^\pi f(t)\sin t dt < f(\frac{\pi}{2}) \Rightarrow \int_0^\pi \sqrt{f(t)} dt < \pi \sqrt{f(\frac{\pi}{2})}$
Assume $f:[0,\pi]\rightarrow (0,+\infty)$ is continuous. And
$$
\int_0^{\pi/2} f(t)\sin t dt =\int_{\pi/2}^\pi f(t)\sin t dt
\tag{1}
$$
if
$$
\int_0^{\pi/2} f(t)\sin t dt < f(\frac{\pi}{2})
\tag{2}
...
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On the proof of the Hölder inequality.
In my attempts of finding different proofs for the Hölder inequality, I have found none like this one that I have derived for myself using the Young inequality: if $p\in]1;\infty[$ and $\frac{1}{p}+\...
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On the validity of certain Gronwall-type inequality
Assume that $u~ \colon \mathbb{R}_+ \to [-M,M]$ is a bounded continuously differentiable function such that $u(0) = 0$ and $$u(t) \leq \int_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{1}\tag{1}$$ where ...
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Issue with a lemma used for proving Morrey's theorem.
I am encountering some issues in proving the following lemma 12.47 from the book "A first course on Sobolev spaces" by Giovanni Leoni.
$\mathbf{Lemma \, 12.47:}$
Take $u \in W^{m,p}(Q_{r}) \...
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Prove $\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$ for reals $\alpha_1,\alpha_2,...\alpha_n$
Prove for all positive integer $n$ and for all $\alpha_1,\alpha_2,...\alpha_n \in \mathbb{R}$ Prove that
$\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$
I have tried by applying ...
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Understanding Proof of Talagrand's Inequality
I am reading Talagrand's seminal paper Concentration of Measure and Isoperimetric Inequalities in Product Spaces. Lemma 2.1.2 on Page 12 obtains the bound
$$
\int_\...
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Upper bound on an integral
Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int_0^1 f(x) dx = 0 $. I would like to know whether the following inequality ...
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a functional inequality (positive functions on circle)
I am trying to prove the following inequality but have no idea at all. Does anyone have any idea how to tackle it or know some related sources?
$$\frac{\int_0^{2\pi}(\frac{1}{2}\frac{f^2}{g}\frac{dg}{...
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Second order differential inequality
Suppose that we have a function $f:[0,T]\to \mathbb R$ in $C^2$, such that for some $C>0,$ $f'' + Cf \geq 0,$ with $f(0)=f'(0)=0.$ This is frequently seen in many places. Is it possible for us to ...
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If $x:[0,1]\rightarrow \mathbb{R}$ verifies $0\leq x(t)\leq \int_0^te^sx(s)^2ds \ \forall t\in [0,1]$ then $x(t)=0 \ \forall t\in[0,1]$
As part of my differential equations homework we have this problem:
If $x:[0,1]\rightarrow \mathbb{R}$ verifies $0\leq x(t)\leq \int_0^te^sx(s)^2ds \ \forall t\in [0,1]$ then $x(t)=0 \ \forall t\in[0,...
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Proving lower bound for an integral
Prop. Let $f\colon [0,1]\times[0,1] \to [0,1]$ be a symmetric measurable function. Then we have $$ \iiint\limits_{[0,1]^3} \prod_{\mathrm{cycl}}f(x,y) \mathrm{d}x\mathrm{d}y\mathrm{d}z +\iiint\limits_{...
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If $g:[a,b] \to \mathbb{R}^m$ is continuous and $x_0 \in [a,b]$, prove $|\int_a^b g(x)dx| \geq \int^a_b|g(x)|dx-2\int^a_b|g(x)-g(x_0)|dx$.
This is an exercise from Leoni’s book on Sobolev spaces. It seems elementary but the proof is eluding me despite my repeated attempts to use the reverse triangle inequality and the mean value theorem. ...
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Proving $\left\vert\int_n^{n+1}s\int_n^xt^{-s-1}dt\,dx\right\vert<\vert sn^{-s-1}\vert$, for $n \in \mathbb{N}$, $x \in [n;n+1]$, $s\in \mathbb{C}$
Let's get out of silence? I try to prove the Riemann hypothesis because I have the humble intuition that it is true, whatever you may think of this step, with my little skills. I have time, so...
I ...
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Bellman-like integral inequality
I have a weakly decreasing, continuous function $w(\cdot)\geq 0$ on $[0,T]$, with $w(0)=1$. I also have a continuous, decreasing function $b(\cdot)$, where $b(t)\geq 0$ for $t\in[0,T]$. I know that
\...
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Integral inequality $\left| \int\limits_{-1}^{1} x f(x) dx \right| \le \left| \int\limits_{-1}^{1} f(x) dx \right|$
I need to find the widest class I can for functions that obey the following inequality:
$$\left| \int\limits_{-1}^{1} x f(x) dx \right| \le \left| \int\limits_{-1}^{1} f(x) dx \right|$$
It is clear ...
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Upper-bound on energy of nonlinear boundary value problem (Hard!)
The problem:
Consider the following boundary-value problem for the function $\rho : \mathbb{R}^{+} \to \mathbb{R}$ with boundary conditions $\lim_{x\to \infty}\rho(x) \to 1$ and $\lim_{x\to 0}\rho(x)...
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if $f'(x)<2f(x)$ and $f(\frac{1}{2})=1$ find the interval in which $\int_{\frac{1}{2}}^1 f(x)dx$ lies
if $f'(x)<2f(x)$ and $f(\frac{1}{2})=1$ and $f$ is a function from $[\frac{1}{2},1]$ to $\mathbb R$ find the interval in which $\int_{\frac{1}{2}}^1 f(x)dx$ lies, given $f(x)$ is non-negative .
...
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Show that an integral is more than $\frac{\left(\frac{\pi}{2}\right)^{r+1}}{r+2}$
Q. Consider the function $$f(r)=\int_0^{\frac{\pi}{2}}x^r\sin(x)dx.$$ Show that $f(r)>\frac{\left(\frac{\pi}{2}\right)^{r+1}}{r+2}.$
I do not know how to even begin. Actually, the first part of ...
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Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.
Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
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Does the $\log$-$\exp$ analogous of Minkowski integral inequality hold true?
Suppose that $X$ is a $\mathcal{X}$-valued random variable and $Y$ is a $\mathcal{Y}$-valued random variable.
Assume that $X$ and $Y$ are independent of each other.
Suppose that $f: \mathcal{X} \times ...
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Energy bound for a closed curve
Let $\gamma : S^1 \rightarrow M$ be a smooth map from a circle of length 1 to a closed manifold $M$ with nonpositive curvature. Could we find a constant $C > 0$ depending only on $M$ such that $$\...
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Upper Bound Product of Integrals
I have an explicit pdf $p(x)$ that I want to show is log-concave, i.e. $\log p(x)$ is a concave function.
Under suitable assumptions, this is equivalent to showing that $p(x)$ satisfies the inequality
...
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Integral inequality $I_2(z) = \mathcal{O}\left(\frac{1}{(1+z)^2}\right)$
I want to prove the inequality of an integral like this:
$$I_2(z) = \int_1^{+\infty} \frac{\rho(t)}{(z+t)^2} d t = \mathcal{O}\left(\frac{1}{(1+z)^2}\right)$$ , when $z \to +\infty$
Below is my ...
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A variant of Cauchy-Schwarz Inequality for integral
I have encountered the above inequality in a proof, where
$u(\tau) \in \mathbb{R}^{n\times m}$, and $w \in \mathbb{R}^n$ is any unit vector
$T_0$ and $t$ is are positive real numbers
The proof ...
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2
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How to bound $m(A)^{-1}\int_A|f(x) - c|^tdx$ above by $\left(m(A)^{-1}\int_A|f(x) - c|dx\right)^t$ for $0 < t < 1$ and $A$ a compact set
Let $m$ denote the Lebesgue measure on $\mathbb{R}^n$, $f$ be a locally integrable function in $\mathbb{R}^n$, let $0 < t < 1$, $A\subset\mathbb{R}^n$ be a compact subset and $c$ some finite ...
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Relationship between bounding volume integral and surface integral
Assume we have a $(d-1)$-dimensional surface $S$ and a sequence of precompact, decreasing open sets $R_{n+1} \subseteq R_{n} \subseteq \mathbb{R}^d$ containing $S$ such that $\cap_{n\in\mathbb{N}} R_n ...
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Minkowski's integral inequality for other norms
In my measure theory course we studied norms $L_p$ and no other norms. For proofs we used exclusively the trick known as Hoelder's inequality which works only on $L_p$ norms. I disliked it very much ...
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Inequality with integral and distribution function
I encountered an inequality with the following variables and functions. $X$ is a random variable drawn from $(-\infty, \infty)$ with cdf $F$, pdf $f$, and mean $\mu=\mathbb{E}[X]$. For any $x$ and $\...
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Suppose $\int_{0}^{a}f(x)dx = 1$. Is it true that for every $b>0$ we have $\int_{0}^{a}\frac{e^{bx}}{b}\left(f(x)\right)^2dx \ge 1 \; ? $
Suppose $a>0$ is such that
$$
\int_{0}^{a}f(x)dx = 1.
$$
Is it true that for every $b>0$ we have
$$
\int_{0}^{a}\frac{e^{bx}}{b}\left(f(x)\right)^2dx \ge 1 \; ?
$$
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Reverse Poincaré inequality and Fourier Transform?
My advisor gave me the other day a draft of a paper he's been working on to study. The following is only a part of a proof which I am trying to figure out at the moment.
Now since, I'm afraid of ...
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an integral inequality involving determinant
I have a matrix of smooth functions ($f_{ij}$) on a compact manifold $M$, and the equality $\int_M det(f_{ij})\ d\omega = 1$; is it possible to get some lower bound on $det(G_{ij})$, where $G_{ij} = \...
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Inequality about integral of probability density function in minimax theory
Recently I have been reading some material about minimax theory and there is an inequality that I don't understand how to derive it. That is
\begin{align*}
\int[p_0^n(x)\land p_1^n(x)]dx \geq\frac{1}{...
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answers
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How to prove this integral inequality by variational method？
Suppose $f(x)$ is fourth differentiable and satisfies $f(0)=f(1)=f'(0)=f'(1)=0$.Prove the inequality$$\frac{1}{a^4}\int^{1}_{0}[f''(x)]^2\text dx\geq \int^{1}_{0}[f(x)]^2\text dx$$Where $a$ is the ...
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Integral Gronwall inequality with negative coefficient
In Wikipedia https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality#Integral_form_for_continuous_functions we can find the following statement:
Let $I$ denote an interval of the real line of the ...
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Question on integral inequality
I am reading the following proof of integral inequality lemma:
Let $b(t)$ and $f(t)$ be continuous functions for $t \geq \alpha$, and $v(t)$ is a differentiable function for $t \geq \alpha$. Suppose ...
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