Questions tagged [holder-inequality]

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

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For $1<p<q<\infty$ show $L^q[0, 1] \subset L^p[0, 1]$ [closed]

For $1<p<q<\infty$ show $L^q[0, 1] \subset L^p[0, 1]$ I know I'm supposed to use holders inequality to solve this, but can't work it out. We haven't learnt anything about measure theory, so ...
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Vectors such that $\lVert x + y \rVert_p^p = \lVert x \rVert_p^p + \lVert y \rVert_p^p$

We fix $1 \leq p < 2$. What are the couple $({x},{y})$ of vectors in $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$) for which the following equality holds \begin{equation} \label{eq:here} \...
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Hölder inequality, showing that if $\int_1^\infty |f(x)| dx$ exists, then exists $\int_1^\infty |f(x)|^2 dx $

So i have a Problem with a proof. given is a function $f:[1,\infty) \to R$, which is in every intervall $[1,b]$ with $b>1$ Riemann-Integratable. There are also given $p,q,r ∈ R$ with $1≤p≤r≤q$. ...
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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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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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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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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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2 answers
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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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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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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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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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Vectorial $L^p$ spaces

Consider the space $$Ł^p(\Omega):=(L^p(\Omega))^N=L^p(\Omega)×L^p(\Omega)×...×L^p(\Omega), \, N \ge 1,\, \Omega \subset \mathbb{R^n},$$ as the vectorial $L^p$ space associated to the scalar one. My ...
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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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$\sum\sqrt{\frac{2a}{b+c}}\le\sqrt[3]{9\sum\frac{a}{b}}$

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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2 votes
1 answer
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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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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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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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3 answers
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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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$f, g$ 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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3 votes
1 answer
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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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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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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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2 answers
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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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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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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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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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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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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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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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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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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, ...
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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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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
2 answers
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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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3 answers
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$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}$ 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'...
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1 answer
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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
2 answers
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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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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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1 answer
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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
3 answers
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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 ...
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1 answer
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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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2 answers
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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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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 ...
1 vote
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Integral version of the Holder Inequality

Hölder Inequality: Let $\left\{a_{1}, a_{2}, \cdots, a_{n} \right\},$ $\left\{b_{1}, b_{2}, \cdots, b_{n}\right\} ,$ $ \cdots\cdots,$ $ \left\{l_{1}, l_{2}, \cdots, l_{n}\right\}$ be $l $ sets of ...
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Hölder's Inequality and $\frac{b}{1/b^\prime} = bb^\prime = \frac{(S_{xy})^2}{S_{xx} S_{yy}}$ get us that $(S_{xy})^2 \le S_{xx} S_{yy}$?

I am currently studying the textbook Statistical Inference by Casella and Berger. Chapter 11.3.1 Least Squares: A Mathematical Solution says the following: For any line $y = c + dx$, the residual sum ...
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4 votes
1 answer
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Holder's Inequality for integrals (non-negative functions)

Let's recall Young's Inequality. Statement: Let $u, v \geqslant 0$, and $p, q \in (0, \infty)$ such that $$ uv \leqslant \frac{u^p}{p} + \frac{v^q}{q}$$ $\blacksquare~$Problem: Let $p,q$ (Holder ...
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1 answer
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Inequalities for generalized means

Given a set of $n$ observations $x_1,\cdots,x_n$, the power mean is defined as $$M_p = \Big(\frac{x_1^p+\cdots + x_n^p}{n}\Big)^{1/p}.$$ Likewise, the exponential mean is defined as $$m_p = \log_p\Big(...

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