Questions tagged [holder-inequality]

Proving or manipulations with inequalities by using Hölder's inequality.

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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 ...
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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} ...
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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), ...
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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} ...
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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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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 ...
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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 ...
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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 $M_p(a)$ ...
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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},...
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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-...
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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 ...
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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 ...
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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(...
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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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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 ...
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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(...
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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/...
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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^{\...
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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? ...
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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 ...
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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 ...
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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 ...
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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 ...
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$||f||_r \leq ||f||_p^{1-c} \cdot ||f||_q^c$ 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^{...
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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 $...
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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'...
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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-...
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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(\...
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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 ...
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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}(...
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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 & ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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\...
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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 $k > 0$, ...
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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 $...
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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 $$\...
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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 ...
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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 ...
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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|...
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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)(...
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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: $...
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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 $\...
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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 ...
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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[...
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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+...

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