Questions tagged [holder-inequality]
Proving or manipulations with inequalities by using Hölder's inequality.
361
questions
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Application of Arzelà-Ascoli theorem
I am trying to understand the wikipedia article on the Arzelà-Ascoli theorem, which can be found here: https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem
More specifically, I am trying ...
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1answer
21 views
Royden 7.2 Exercise 22: Inequality for the Indefinite Integral of an $L^p[a, b]$ function
The following is Exercise 22 from Royden $\S$7.2:
For $1 \leq p < \infty$, if the absolutely continuous function $F : [a, b] \to \mathbb{R}$ is the indefinite integral of an $L^p[a, b]$ function $...
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Application of Holder's inequality for proving convergence [closed]
How do I show that if $$\lim_{n\to \infty}\int^1_0|f_n(x)|^3dx=0$$
$$\lim_{n\to\infty}\int^1_0\frac{f_n(x)}{\sqrt{x}}dx=0?$$
I was thinking of using the Holder's inequality, but I'm stuck... I think ...
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1answer
22 views
Problem using Hölder inequality
Suppose we have $(X, \mathcal A, \mu )$ a measure space with $\mu (X)=1$ and we have measurable functions $f,g:X \rightarrow (0, \infty ]$ with $f(x)g(x)\geq 1 $ for all $x \in X $.
How do I show that ...
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29 views
Seemingly Simple Probabilistic Inequality Question
Let's assume all random variables here take positive values. All RVs will be represented in capital letters and constants in lower case. First, we know that
$$\big| \mathbb{E}[A] - \mathbb{E}[B] \big| ...
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2answers
40 views
Hölder's inequality with $p=1,q=\infty$ and SDEs
I have an expression of an expectation of stochastic processes at time $T$, $$\mathbb{E}[X_TM_T]$$
$M_T$ is a martingale such that $\mathbb{E}[M_T] = \mathbb{E}[M_0] = M_0 = 1 $. I am tempted to ...
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Holder-type inequality or not (optimal transport in Riemannian manifolds)?
I'm reading a paper about optimal transport on Riemannian manifolds, and I came into a part where the author talks about "distortion coefficients". Here there's an inequality that I don't ...
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An inequality for Bochner functions
Let $\Omega\subset \mathbb{R}^n$ be bounded and open, $0<T<\infty$. Assume $f,g\in L^{\infty}((0,T),H_0^1(\Omega))\cap L^2((0,T),H^2(\Omega))$ and $f',g'\in L^2((0,T),L^2(\Omega))$. I want to ...
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An application of the Hölder's inequality to Young's inequality
Hölder's inequality Let $f_1, f_2,\dots, f_k\colon X\to \overline{\mathbb{R}}$ be a measurable functions and $p,p_1,\dots, p_k\in [1,\infty)$ such that $$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots ...
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Corollary Hölder's inequality: $\Vert f \Vert_p=\sup_{\{g\::\:\Vert g\Vert _q=1\}}\int fg$
I want to prove the following corollary of Hölder's inequality:
$f\in L^p \Longrightarrow \Vert f \Vert_p=\sup_{\{g\::\:\Vert g\Vert _q=1\}}\int fg$
where $\frac{1}{p}+\frac{1}{q}=1$. I have already ...
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1answer
38 views
Intuitive implication of the fact that dual of $L_{p}$-norm space is $L_{q}$-norm space where $\frac{1}{p}+\frac{1}{q}=1$.
While studying the inverse problem theory (I am mainly concerning discrete variables), I learned the theorem that "the dual of $L_{p}$ where $1<p<\infty$ is $L_{q}$ provided that $\frac{1}{...
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Prove that $\cup_{1<p\leq\infty} L^p([0,1])\subset L^1([0,1]) $.
Prove that $\cup_{1<p\leq\infty} L^p([0,1])\subset L^1([0,1]) $.
Let $f\in\cup_{1<p\leq\infty}L^p([0,1])$.
Then $f\in L^p([0,1])$ for some $p$. We write $\int|f|d\mu < (\int|f|^p\ )^\frac{1}{...
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2answers
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Application of Hölder's inequality: $(a+b)^t \le 2^t(a^t+b^t)$ for $t\ge 1.$
While searching for a proof of the algebra property of Sobolev spaces ($H^s(\mathbb R^n)$ is an algebra when $s > n/2$), I found these notes. On page two the author states that if $t \ge 1$ then ...
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1answer
40 views
Vectorial $L^p$ spaces
Let $\Omega \subset \mathbb{R^n}$
Denote:
$Ł^p(\Omega):=(L^p(\Omega))^N=L^p(\Omega)×L^p(\Omega)×...×L^p(\Omega)$, $N$ times, as the vectorial $L^p$ space associated to the scalar one, where $N \ge 1$.
...
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1answer
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$f^{p_1}$ belongs to $L^p(E)$ and $g = \chi_{E}$ belongs to $L^q(E)$?
I was reading a Corollary in Royden, which states:
In the proof,
The book mentioned by observation that $f^{p_1}$ belongs to $L^p(E)$ and $g = \chi_{E}$ belongs to $L^q(E)$. I am not sure how this ...
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0answers
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$\sum\sqrt{\frac{2a}{b+c}}\le\sqrt[3]{9\sum\frac{a}{b}}$
Let $a$, $b$ and $c$ be positive numbers. Prove that:
$$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}}\le\sqrt[3]{9\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$$
It is from ...
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When can we deduce $(mA+nB+pC)>(mX+nY+pZ)$ from $(A+B+C)>(X+Y+Z)$?
I am trying to prove an inequality in the form $(mA+nB+pC)>(mX+nY+pZ)$.
I can prove the inequality $(A+B+C)>(X+Y+Z)$ and I wonder if there is any condition,
under which we can deduce $(mA+nB+...
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1answer
9 views
An interpolation inequality using Holder inequality
If $f \in L^{p}\left(\mathbb{R}^{n}\right) \cap L^{q}\left(\mathbb{R}^{n}\right),$ where $p<q,$ prove that $\|f\|_{r} \leq\left(\|f\|_{p}\right)^{\frac{1 / r-1 / q}{1 / p-1 / q}}\left(\|f\|_{q}\...
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1answer
40 views
Kind of equality on the sphere in Holder inequality
Lets suppose p > 1 and q as $\frac{1}{p} + \frac{1}{q} = 1$
$g$ is function of $L^q(\Omega, A, \mu) $
We suppose that exists $M$ positive number as : $M = sup\left \{ \int fg d\mu ; f \in \...
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1answer
15 views
Let $f, g :(0,1) \to \mathbb{R}\,\,$ s.t. $\lVert g \rVert_2 =3$ and $\lVert f \rVert_1 = e\,\,$, show that $|\int_{0}^{1} g \sqrt{\log(f)}dx| \le 3$
Let $g\in L^2((0,1))$ s.t. $\lVert g \rVert_2 =3\,\,\,$ and $f \ge1, f \in L^1((0,1))$ s.t. $\lVert f \rVert_1 = e\,\,$, prove that:
$$ \Biggl|\int_{0}^{1} g \sqrt{\log(f)}\,dx \Biggr|\,\,\le\,\,3$$
...
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2answers
130 views
Algebraic inequality $\sum \frac{x^3}{(x+y)(x+z)(x+t)}\geq \frac{1}{2}$
The inequality is
$$\frac{x^3}{(x+y)(x+z)(x+t)}+\frac{y^3}{(y+x)(y+z)(y+t)}+\frac{z^3}{(z+x)(z+y)(z+t)}+\frac{t^3}{(t+x)(t+y)(t+z)}\geq \frac{1}{2},$$
for $x,y,z,t>0$.
It originates from a 3-D ...
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1answer
20 views
$f, g$ measurable, $\lVert g\rVert_1=1$ and $0 \le a \le f \le b$ so $\int_{\mathbb{R^n}} \frac{g}{f}dx \ge \frac{1}{\int_{\mathbb{R^n}} fg dx}$
Let $g \ge 0$ be measurable in $\mathbb{R^n}$ s.t. $\lVert g\rVert_1=1$ and $f$ measurable s.t. $a \le f \le b$, for $a, b \gt0$, show that: $$\int_{\mathbb{R^n}} \frac{g}{f}dx \ge \frac{1}{\int_{\...
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1answer
64 views
How is the Prékopa Leindler inequality different from Holder's inequality?
The following is an extract from these notes (Pg. 30), and I'm not able to understand the part towards the end. I would appreciate any help!
Prékopa-Leindler Inequality: Let $f$, $g$ and $m$ be non-...
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1answer
67 views
Proof of Hölder's inequality by weighted AM-GM inequality
I am going throw my notes, and for the proof of Hölder's inequality, I wrote the following:
Theorem. (Hölder) Let $1 \leq q,r \leq \infty $ such that $ \dfrac{1}{q} + \dfrac{1}{r} = 1 $ (here we ...
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1answer
59 views
How to prove the inequality $\prod_{k=1}^n ((1-\beta)+\beta r_k)\le\left((1-\beta)+\beta\sqrt[n]{\prod_{k=1}^n r_k}\right)^n$
Let $\beta\geq1$, $n\in\mathbb N$ and $r_1,\dots, r_n\geq1-\frac1\beta$ be real numbers.
I want to prove that
Claim. $$\prod_{k=1}^n (1+\beta(r_k-1))\le\left(1+\beta\left(\sqrt[n]{\prod_{k=1}^n r_k}-...
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2answers
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The convergence of $\int_0^{\infty} \sin(x)(x-[x])/x^{\alpha}$
I need to evaluate the following integral
$$\int_0^{\infty} \sin(x)(x-[x])/x^{\alpha}dx$$
where $\alpha\in (0,1)$ and $[x]$ is the floor function.
Without $x-[x]$, we can evaluate the integral easily ...
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1answer
85 views
Proving $\sum_{cyc}\sqrt[3]{\frac{1}{a}+\frac{2}{bc}+a+2b+c}\leq\frac{6}{abc}$ for positive values such that $ab+bc+ca=3$
This problem is from my teacher.
Know that $a,b,c>0$, $ab+bc+ca=3$.
Prove that: $$\sum_{cyc}\sqrt[3]{\frac{1}{a}+\frac{2}{bc}+a+2b+c}\leq\frac{6}{abc}$$
I tried to change the number '$6$' into $...
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votes
1answer
55 views
Norm inequality of Fourier transform
Let $f\in L^1(R^d)$, where
$$\hat{f}(\xi) = \int f(x) e^{-2\pi\xi i x}dx$$
and
$$\hat{f} = f$$
How may we prove
$$\|\hat{f}\|_p \leq \|f\|_1$$
holds?
I tried to use Holder's inequality, but end up in ...
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0answers
24 views
Hölder's inequality aplication
I have to do this exercise:
Let $f\in{L^{p}([0,1])}$, with $1<p<+\infty$. Prove that $\lim\limits_{j\to\infty} {j^{(2p-2)/p}}\left| \int_{1/(j+1)}^{1/j} {f}\mathrm{d}m\right\|
=0$.
I don't know ...
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1answer
87 views
a question about $L^p$ space and Holder inequality
let $(X,A,\mu)$ a measurable space.$1<p, q, s<\infty$
a.Prove that if $f\in L^p$ and $g\in L^q$ such that $1/p+1/q=1/s$ then $fg\in L^s$ and $\|fg\|_s\leq \|f\|_p\|g\|_q$.
b.Prove that if $f\in ...
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1answer
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Application of Hölder to prove $L^q \subset L^p$ for $1\leq p\leq q<\infty$ with Lebesgue measure
I'm working on the following problem and got stuck. Any help would be really appreaciated.
Let $a,b\in\mathbb{R}$ and $1\leq p \leq q < \infty$. Show that for any $f\in L^q([a,b])$ the following ...
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1answer
68 views
Applying Hölder's inequality to convolutions [closed]
In Adam's Sobolev Spaces, he claimed that the following inequality follows from Hölder's inequality
$$|J_\epsilon*u| := \left|\int_{\mathbb{R}^n} J_\epsilon(x-y)u(y)dy\right| \leq \left(\int_{\mathbb{...
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2answers
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prove thatt $\sum_{cyc} \frac{1}{{(x+y)}^2}\ge 9/4$
prove that $$\sum_{cyc} \frac{1}{{(x+y)}^2}\ge 9/4$$ where $x,y,z$ are positives such that $xy+yz+xz=1$
By Holder;$$\left(\sum_{cyc} \frac{1}{{(x+y)}^2} \right){\left(\sum yz+zx \right)}^2\ge {\sum \...
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Computing using holder's and young's inequality
I need help in obtaining the following estimate which am stuck in calculating it in details:
$$
\big| ((v^2+2v\langle u\rangle-v, (-\Delta)^{-1}v))\big|
$$
$$
\leqslant \dfrac{3c_0}{8}\int_{\Omega}(v^...
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0answers
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Showing that $\|f\|_{p}=\sup\left\{ \left|\int_{X}fg\right\|:g\in L^q(X),\|g\|_{q}\leq 1\right\}$ [duplicate]
Let $f\in L^p(X)$ and $1/p+1/q=1$. Show that $\|f\|_{p}=\sup\left\{ \left|\int_{X}fg\right\|:g\in L^q(X),\|g\|_{q}\leq 1\right\}$
Hi. I'm stuck proving that $\|f\|_{p}$ is the lowest upper bound.
I ...
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0answers
23 views
Applying Holders Inequality with supremums
I'm reading through some statistics proofs and have a question:
Holder's inequality states for some vector norm $|\cdot|$ (at least in my notes): $E[|XY|]\leq (E[|X|^{1/a}])^a (E[|X|^{1/b}])^b$
Let $a=...
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50 views
discrete Hausdorff-Young inequality
I am consider inequality about hausdorff young inequality on group,the group is $\mathbb{Z}(n)$,we know Hausdorff Young inequality such like $ \left\|f_1*f_2\right\|_r\leq C\left\|f_1\right\|_{q_1} \...
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2answers
225 views
Generalizing Titu's Lemma
I found a nice generalization of Titu's Lemma and was wondering if this has a name or has a reference anywhere.
Let $m$ be an integer greater than or equal to 2, $a_i^m$ a non-negative real number, ...
0
votes
1answer
35 views
Three doubts in the proof of Hölder's inequality
I quote Jacod-Protter (2004).
Theorem (Hölder's inequality) Let $X$, $Y$ be random variables with $\mathbb{E}\{|X|^p\}<\infty$, $\mathbb{E}\{|Y|^q\}<\infty$, where $p>1$ and $\frac{1}{p}+\...
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1answer
73 views
For any real positive numbers $a, b, c$, prove that $3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2) \geq abc(a+b+c)^3$ [duplicate]
My progress is that I applied Hölder’s for this,
$3(a^2b+b^2c+c^2a) \frac{(ab^2+bc^2+ca^2)}{abc} \geq (a+b+c)^3$
whereas $3(a^2b+b^2c+c^2a) \frac{(ab^2+bc^2+ca^2)}{abc} = (1+1+1)(a^2b+b^2c+c^2a)(\...
2
votes
2answers
97 views
Product of averages
Let $(x_1,...,x_n)$ and $(y_1,...,y_n)$ be two different tuples of positive reals such that $x_1\times\dots\times x_n=y_1\times\dots\times y_n = c$.
Is it true that
$$\left(\frac{x_1+y_1}{2}\right)\...
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votes
3answers
84 views
$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}$ for $a,b,c\in\mathbb{R}^+$ with $abc=1$
Suppose that $a,b,c$ are positive reals such that $abc=1$. Prove that $$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}.$$
Hint: Use Titu's lemma.
My approach: I am trying to use Titu'...
0
votes
1answer
108 views
Holder inequality of Schatten norm for p=q=2 and Hermitian operators
I was reading some Linear Algebra contents and then I encountered this problem, I tried to prove this by using for example Holder inequality for Schatten norms, But I didn't succeed. The question is :
...
5
votes
2answers
112 views
Show that $\int_0^1 f^3(x) dx + \frac{4}{27} \ge \left( \int_0^1 f(x) dx \right)^2$, where $f',f'' >0$
Let $f :[0,1] \to [0,\infty)$, $f$ is twice differentiable, $f'(x) >0$, $f''(x) >0$ for all $x\in [0,1]$. Prove that
$$ \int_0^1 f^3(x) dx + \frac{4}{27} \ge \left( \int_0^1 f(x) dx \right)^2.$$
...
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votes
1answer
47 views
Holder's Inequality application
Given two sequence $\{x_{k}\}_{k\ge 1}, \{y_{k}\}_{k\ge 1},$ Holder's inequality states that
$$\sum_{k=1}^{\infty} |x_k\,y_k| \leq \biggl( \sum_{k=1}^{\infty} |x_k|^2 \biggr)^{\frac{1}{2}} \biggl( \...
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1answer
79 views
Showing Holder's inequality holds for $p=\infty$ and $q=1$
We're asked to show that Holder's inequality (for the case when $1/p + 1/q = 1$) holds for the case when $p=\infty$ and $q=1$. The inequality is given to us in the following form.
$\sum\limits_{i=0}^\...
2
votes
3answers
60 views
Inequality with a, b, c about finding minimal and maximal value
Find the minimal and maximal value (if they exist) of ${\sqrt{\frac{a(b+c)}{b^2+c^2}}} +{\sqrt{\frac{b(a+c)}{a^2+c^2}}} +{\sqrt{\frac{c(b+a)}{b^2+a^2}}}$ if are non-negative real numbers, such that ...
0
votes
1answer
46 views
Pseudo Otto Holder proof help.
I'm at the first part of a pseudo Otto Holder theorem which claims if $1/p + 1/q = 1$ where $x \in \ell_p$ and $a \in \ell_q$ then $\sum_\limits{i=0}^\infty |a_i x_i| \leq ||a||_q||x||_p$.
For the ...
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votes
2answers
369 views
Prove that $\left(x + \sqrt[3]{abc}\right)^3 \le (x + a)(x + b)(x + c) \le \left( x + \frac{a + b + c}{3} \right)^3.$
Let $x,$ $a,$ $b,$ $c$ be nonnegative real numbers. Prove that $$\left(x + \sqrt[3]{abc}\right)^3 \le (x + a)(x + b)(x + c) \le \left( x + \frac{a + b + c}{3} \right)^3.$$
I've tried using AM-GM by ...
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1answer
58 views
prove that $\sum_{cyc}\frac{a^2}{b}\ge 4$ [duplicate]
prove that $$\sum_{cyc}\frac{a^2}{b}\ge 4$$ if $a,b,c,d\ge 0$,$a^2+b^2+c^2+d^2=4$
My try:
by Titu's lemma
$$\sum_{cyc}\frac{a^2}{b}\ge a+b+c+d$$
now from given condition we can say $a\ge 0,a\le 2$,,or ...
