Questions tagged [holder-inequality]
Proving or manipulations with inequalities by using Hölder's inequality.
461
questions
0
votes
1
answer
34
views
Applying Holder Inequality to obtain a estimate.
I'm reading a paper concerned with PDE, a inequality in it makes me confuse.
We have known:
$$\int|\nabla u|^{p-2}|A|^2\eta^2\leq C\int|\nabla u|^{p}$$
and author says we can obtain following ...
- 33
0
votes
1
answer
24
views
Young's inequality in probability theory
The following is the standard version, in two equivalent statements:
If $a \geq 0$ and $b \geq 0$ are nonnegative real numbers and if $p > 1$ and $q > 1$ are real numbers such that $\frac{1}{p} ...
- 39
0
votes
0
answers
36
views
Holder's Inequality for p < 0 or q < 0
We have the theorem that:
If $u_k,v_k$ are positive real numbers for $k = 1,...,n$ and $\frac{1}{p} + \frac{1}{q} = 1$ with real numbers p and q, such that pq < 0 (i.e. p < 0 or q < 0), ...
0
votes
1
answer
58
views
How to use Hölder inequality to prove this integral inequality?
Consider an integral operator $Tf(x)=\int_{\mathbf{R}^n}K(x,y)f(y)dy.$ And $s,r \in(0,\infty), s \geq r$ are two indices. I would like to prove
\begin{equation}
\| Tf\|_{r} \leq (\int_{\mathbf{R}^n} ...
1
vote
1
answer
54
views
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 ...
- 5,343
0
votes
3
answers
96
views
I want to prove for all $a > 0 $, $b > 0$, $\frac{2ab}{a + b} + \sqrt{\frac{1}{2}\left(a^2 + b^2\right)} \geq \frac{a + b}{2} + \sqrt{ab}$ [closed]
How can I prove that for all $a > 0$ and $b > 0$, $$\frac{2ab}{a + b} + \sqrt{\frac{1}{2}\left(a^2 + b^2\right)} \geq \frac{a + b}{2} + \sqrt{ab}$$
I found this inequality in an mathematic ...
- 75
2
votes
1
answer
94
views
Pulling out a maximum term from an integral
Suppose $g$ is nonnegative on $[0,1]$. Is the following true?
$$ \left| \int_0^1 f(x) g(x) dx\right| \leq \|g\|_\infty \left| \int_0^1 f(x) dx\right| $$
This would be like Holder's inequality, except ...
- 1,716
0
votes
1
answer
48
views
Bounding growth of generalised mean $M_p(a)$ as $p$ increases
This is a follow-up question to the one previously asked here. on the geometric growth rate of generalized mean. For some $p$ and positive numbers $a=(a_1,\dots,a_n)\in\mathbb{R}^{+}$, let $M_p(a)$ ...
- 627
3
votes
1
answer
53
views
relationship between generalized mean ratios $M_4(a)/M_2(a)$ vs $M_2(a)/M_1(a)$
For some $p$ and positive numbers $a=(a_1,\dots,a_n)\in\mathbb{R}^{+}$, let $M_p(a)$ denote the generalised or Hölder mean, defined as
\begin{align}
M_p = \left(\frac1n\sum_{i=1}^n a_i^p \right)^{1/p},...
- 627
0
votes
0
answers
8
views
Hölder kind bound for adjungated operators
I have an linear, compact and selfadjungated operator L on some Hilbertspace H with eigenvalues between zero and one. Let $v_i\in H$.
Does the following bound somehow hold? :
$$
\left\|\sum_{j=1}^n(I-...
- 145
2
votes
1
answer
63
views
show inequality with Hölder or induction?
For $a_j\in\mathbb R$ with $a_0=0$ show that $\sum_{j=1}^na_j^2\leq n^2\sum_{j=0}^{n-1}(a_{j+1}-a_{j})^2$.
First I tried to use induction but this doesn't work. Then I tried to use the Hölder ...
- 23
1
vote
0
answers
25
views
Are there any refinements of $\langle \vec x,\vec y\rangle \leq \lVert \vec x\rVert_1\lVert \vec y\rVert_\infty$?
The standard Young's inequality states, for $a,b>0$ and $\nu\in[0,1]$, that
$$
\nu a + (1-\nu)b\geq a^{\nu}b^{1-\nu}.
$$
This has been refined in various ways, for example see the refined Young's ...
- 6,834
3
votes
1
answer
86
views
About functions in fractional Sobolev spaces $W^{s,p}(\mathbb{R})$ and $\lim_{x\rightarrow a^{+}}\int_{a}^{x}\frac{f(y)}{(x-y)^{s}}dy=0$
Let $1\leq p<+\infty$, $0<s<1$ and $f\in W^{s,p}\left(\mathbb{R}\right)$, where $$W^{s,p}\left(\mathbb{R}\right):=\left\{ u\in L^{p}\left(\mathbb{R}\right):\frac{\left|u\left(x\right)-u\left(...
- 287
0
votes
2
answers
25
views
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 ...
- 413
0
votes
0
answers
47
views
Intuition for proofs of Minkowski Inequality
I've always found the usual proofs of both Minkowski's and Hölder's inequalities extremely unintuitive and unsatisfying. It is very unclear how someone who wanted to prove Minkowski's inequality would ...
- 149
0
votes
1
answer
27
views
show that $g \in L_2((1, \infty))$ defined by $g(x):=\frac{1}{x} \int_{[x,10x]} \frac{f(t)}{t^{\frac{1}{4}}}dλ_1(t).$
I have the following problem:
let $1 < p< 4$ and let $f \in L_p((1, \infty))$.
We consider $g: (1, \infty ) \rightarrow [- \infty, \infty]$ , defined by $g(x):=\frac{1}{x} \int_{[x,10x]} \frac{f(...
- 388
1
vote
1
answer
33
views
Hölder-like inequality $\|f\|_\pi\|g\|_\infty\le\frac{1}{16}\|f\|_\pi^2+4\|g\|_\infty^2$
I don't know how the second bound occurs. It's not Hölder, Cauchy-Schwarz or Minkowsi.
Of course we have:
It appears in the following paper on page 531:
https://projecteuclid.org/journals/bernoulli/...
- 65
1
vote
1
answer
64
views
Holder's inequality on weighted $L^p$ space
Let $v(x,z)$ be a function, where $x\in\mathbb{R}^d$ is a $d$-dimensional real vector and $z\in(0,\infty)$. Let $D\subset \mathbb{R}^d$ be a domain and $|\alpha|<1$. Consider the space $L^2(z^{\...
- 141
0
votes
1
answer
51
views
Help with Holder's Inequality [closed]
I was studying Holder's inequality and I came across the second problem used at the 2001 IMO because it involved Holder. The question I want to ask is what is the reasoning behind the first line? ...
- 39
1
vote
0
answers
27
views
For 1≤p<q≤$\infty$, show that $L^{p}(\mathbb{Z},\mu) \subset L^{q}(\mathbb{Z},\mu)$, $\mu$ being the counting measure. [duplicate]
I have to show the inclusion $L^{p}(\mathbb{Z},\mu) \subset L^{q}(\mathbb{Z},\mu)$ with the counting measure. Using 1≤p<q≤$\infty$ and the definition of the $L^{p}/L^{q}$ spaces I have tried to ...
- 67
1
vote
3
answers
83
views
For $x,y,z∈ℝ^{+}$,without using Hölder's inequality prove that $\frac{x}{\sqrt{x^2+8yz}}+\frac{y}{\sqrt{y^2+8xz}}+\frac{z}{\sqrt{z^2+8xy}}\geq1$.
For $x,y,z∈ℝ^{+}$, prove that $\frac{x}{\sqrt{x^2+8yz}}+\frac{y}{\sqrt{y^2+8xz}}+\frac{z}{\sqrt{z^2+8xy}}\geq1$.
In this question solution used Hölder's inequality, but I am looking a solution ...
user1135351
0
votes
1
answer
50
views
Hölder Condition and equicontinuity
My question is the following: If $|f_n(x)-f_n(y)|≤M|x-y|^\alpha$ for some fixed M and $\alpha>0$ and all x,y in a compact interval, show that ${f_n}$ is uniformly equicontinuous.
As a criterion for ...
- 81
0
votes
2
answers
108
views
Proof of Holder's Inequality in Multivariable Calculus
I am self studying Calculus 3 (Multivariate/Vector Calculus) as a hobby and I came across this question. I graduated with a degree in chemistry, so I know a bit of math, but not too much to answer ...
- 9
0
votes
0
answers
21
views
$||f||_r \leq ||f||_p^{1-c} \cdot ||f||_q^c$ from Hölder inequality
I need some help with the following problem
Let $1 < p < q <\infty$, $f \in L^p(\mu)\cap L^p(\mu)$ and $r \in [p,q]$. Show that there exists a $c \in [0,1]$ such that
$$||f||_r \leq ||f||_p^{...
- 352
0
votes
0
answers
30
views
Clarifications on the assumptions of Poincaré inequality
I am following the proof of the following Poincaré's inequality from the Lieb and Loss textbook
8.12 Let $Ω∈\mathbb{R}^n$ be a bounded, connected open set that has the cone property for some $θ$ and $...
- 83
2
votes
2
answers
45
views
Convergence of the product of an $L_1$-convergent function f and a point-wise a.e. convergent function g.
Let $f_k,f$ be integrable functions with $f_k\to f$ in $L_1$, $g_k,g$ are measurable with $\sup_k\|g_k\|_{L_\infty}<\infty$ and $g_k\to g$ p.w. a.e.. Show that $f_kg_k\to fg$ in $L_1$.
Using Hölder'...
- 21
0
votes
0
answers
17
views
Proving $||u||_b \leq ||u||_a^\lambda||u||_c^{1-\lambda}$ [duplicate]
Here is an interpolation inequality of the norms: if $a\leq b\leq c$ and suppose $\lambda$ satisfies $$\frac{1}{b}=\frac{\lambda}{a}+\frac{1-\lambda}{c}$$
then $$||u||_b \leq ||u||_a^\lambda||u||_c^{1-...
- 53
0
votes
0
answers
35
views
Hölder's inequality for two variables
I have one question about the Hölder's inequality.
Let $f:\Omega \times [0,\infty) \rightarrow \mathbb{R}$ be in $L^{\infty}(\Omega\times[0,\infty))$ and $g: \Omega \rightarrow \mathbb{R}$ be in $L^1(\...
- 40
6
votes
1
answer
340
views
Prove that: $\sqrt{\frac{4x^2+y^2}{3x^2+yz}}+\sqrt{\frac{4y^2+z^2}{3y^2+xz}}+\sqrt{\frac{4z^2+x^2}{3z^2+xy}}\ge\frac{3\sqrt{5}}{2}$
Let $x$, $y$ and $z$ be positive numbers. Prove that:
$$\sqrt{\frac{4x^2+y^2}{3x^2+yz}}+\sqrt{\frac{4y^2+z^2}{3y^2+xz}}+\sqrt{\frac{4z^2+x^2}{3z^2+xy}}\ge\frac{3\sqrt{5}}{2}.$$
This problem is ...
1
vote
1
answer
56
views
Strict version of Hölder's inequality when $p=1$ and $q=\infty$
How do I prove that there is no function $h : \mathbb{R} \to \mathbb{R}$ of unit norm on $L^1$ such that
$\int_{\mathbb{R}} f h = \lVert f \rVert_{\infty} $
Whenever $f$ is a function (like $\tan^{-1}(...
- 13
0
votes
1
answer
33
views
Well definedness & boundedness of operators
Are the following linear operators well defined bounded operators $\ell^2 \to \ell^1$: $$\begin{align*} T: (x_k)_{k=1}^\infty &\mapsto (k^{-1}x_{k+2})_{k=1}^\infty \\\\ L:(x_k)_{k=1}^\infty & ...
- 109
0
votes
0
answers
31
views
Bounding integral from individual bounds
I am trying to bound the following integral: $\int\limits_{0}^a f(x) g(x)\ dx$ where
$-\delta \leq f(x),g(x) \leq 1+\delta$ for all $x$.
I know that individual integrations are vanishing. That is, for ...
- 245
0
votes
1
answer
52
views
Generalizing Cauchy-Schwarz inequality and Hölder's inequality for concave functions
We know that $$E[x^2] > E[x]^2$$ by Cauchy-Schwarz. Further we know that $$E[x^2]E[\sqrt{x}] > E[x]E[x^{1.5}]$$ when $x$ is restricted to positive reals by Callebaut's Inequality or Hölder's ...
- 385
2
votes
1
answer
27
views
Is $e^{\gamma t}$ Hölder continuous, with $\gamma<0$.
Is $e^{\gamma t}$ Hölder continuous?, with $\gamma<0$.
This question appear in something that i am working, i am not sure if the answer is yes or not.
My only attempt is for definition $|e^{\gamma ...
- 1,115
3
votes
1
answer
71
views
How to use Holder's inequality in proving convexity of sum of exponentials
Let $f(\mathbf{x})=\sum_{k=1}^{n}e^{x_{k}}=e^{x_{1}}+e^{x_{2}}+\ldots+e^{x_{n}}$ where $x:=\begin{bmatrix} x_{1}&\cdots&x_{n}\end{bmatrix}^{T}$
I want to prove that $f$ is convex. I thought ...
- 398
1
vote
1
answer
61
views
An estimate on the trace norm of a product of operators
I am having some troubles understanding the following paper of Demuth, Stollmann, Stolz and Van Casteren that improves the Hölder inequality for the trace norm : https://link.springer.com/article/10....
- 165
0
votes
2
answers
95
views
Deduce the Hölder Inequality
If $f, g$ are two functions $\geq 0$ in an interval $I$ such that the
integrals $\int_I f(t) d t$ and $\int_I g(t) d t$ are convergent, the
integral $\int_I(f(t))^\alpha(g(t))^{1-\alpha} d t$ is ...
- 217
2
votes
0
answers
66
views
Bounding the expectation of a product of random variables using Hölder's inequality
Let $X_1, X_2, \ldots, X_k$ be centered random variables which are not necessarily independent. Under what conditions do we have a bound of the form $$\mathbb{E} \left\lvert X_1 X_2 \cdots X_k \right\...
- 10.3k
3
votes
1
answer
79
views
Product of power means inequality: prove $M_0^{\frac{n}{n+k}}M_k^{\frac{k}{n+k}} \leq M_1$
Inspired by this question Prove that $\frac{{x_1}^2+{x_2}^2+\cdots+{x_n}^2}{n}x_1x_2\cdots x_n\le\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)^{n+2}$, I wonder if the following is true:
Let $k > 0$, ...
- 2,001
1
vote
1
answer
39
views
Linear operator on sequences $M_y$ given by $M_y(x) = (y_jx_j)_{j \in \mathbb{N}}$ preserves $l^p$ iff $y \in l^{\infty}$
Let $y = (y_j)_{j \in \mathbb{N}} \subset \mathbb{K}$ where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ be a fixed sequence. For each sequence $x = (x_j)_{j \in \mathbb{N}}$ define a new sequence $...
- 717
1
vote
1
answer
137
views
Equality in Holder's inequality if $(|x_i|/\|x\|_p)^{1/q}=(|y_i|/\|y\|_q)^{1/p}$
Holder's inequality states that $\langle x, y\rangle \le \langle u, v\rangle \le \| x\|_p \|y \|_q$ where $u_i=|x_i|,v_i=|y_i|$ and $p+q=pq,p\ge 1$. We observe that equality occurs if and only if
$$\...
- 10.7k
1
vote
0
answers
55
views
Derivation of Hölder Inequality through Young's Inequaliy
I am having trouble following a proof where Young's Inequality is being used to derive Hölder's Inequality. More precisely, there is a particular and final step that utilizes integration in order to ...
1
vote
1
answer
87
views
Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus
Note: this has been edited to make the question more general.
I want to show that $(1+x)^k/k + (1-x)^m/m$ is minimized at $x=0$ when $k,m\geq 1$ and $-1\leq x \leq1$.
Of course, I could take the first ...
- 234
1
vote
0
answers
60
views
Proving the uniform convexity of $L^p$ for $1 < p \le 2$
We are asked to show the uniform convexity of $L^p$ for $1 < p \le 2$ using the following inequality:
For all $1 < p < \infty$, there is a constant $C$ such that $|a - b|^p \leq C(|a|^p + |b|...
- 21
1
vote
1
answer
183
views
Wasserstein metric vs Holder continuity
It is well known that if $f$ is a Lipschitz continuous function, i.e.
$$\forall x,y\in \Omega\qquad |f(x)-f(y)|\le L\|x-y\|$$
then, for any two probability distributions $\mu, \nu$
$$\int_\Omega f(x)(...
- 1,139
0
votes
2
answers
41
views
Deduction of Hölder Inequality
If the Hölder inequality holds, we have
$$
|x\cdot y|\leq \| x\|_p\| y\|_q
$$
now if $y\neq 0$ this leads to
$$
\frac{|x\cdot y|}{\| y\|_q}\leq \| x\|_p
$$
Now my question.
Is this implication true:
$...
- 1,650
0
votes
0
answers
35
views
Inclusion of Holder Spaces.
From the wikipedia page on Holder spaces it says that if $0 < \alpha < \beta \leq 1$, then there is an inclusion map $\iota : C^{0 , \beta, }(\Omega) \rightarrow C^{0 , \alpha}(\Omega)$, where $\...
- 139
0
votes
0
answers
16
views
Founding bounds on a certain expression
Suppose we have a bounded domain $\Omega$ with a boundary $\Gamma$. The space $L^2(\Omega)$ is equipped with the usual norm and inner product $|| \cdot||$ and $(\cdot , \cdot)$.
I was working on a ...
- 156
1
vote
1
answer
234
views
Inequality of expectation of product [closed]
Suppose non-negative random variables $X_1,X_2,\ldots,X_N$, and they are maybe dependent. Is the following inequality correct?
\begin{align}
E\left[\prod_{n=1}^{N} X_n^{k_n}\right] \le \max_n E\left[...
- 597
0
votes
1
answer
104
views
Prove that $\frac{x^2}{x^2+2y}+\frac{y^2}{y^2+2z}+\frac{z^2}{z^2+2x} \ge 1$
Prove that
$$\frac{x^2}{x^2+2y}+\frac{y^2}{y^2+2z}+\frac{z^2}{z^2+2x} \ge 1$$
with $xy+yz+zx=3$ and $x,y,z >0$.
If I use Cauchy-Schwarz then $\frac{x^2}{x^2+2y}+\frac{y^2}{y^2+2z}+\frac{z^2}{z^2+...
- 33