Questions tagged [holder-inequality]

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

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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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43 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}(...
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1 answer
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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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175 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[...
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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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Find the least constant $C$ such that $\sum_{1\leq i<j\leq n}x_ix_j(x_i^2+x_j^2) \leq C(\sum_{i=1}^nx_i)^4$

Find the least constant $C$ such that for all nonnegative real numbers $x_1,x_2,.......,x_n$ $$\sum_{1\leq i<j\leq n}x_ix_j(x_i^2+x_j^2) \leq C(\sum_{i=1}^nx_i)^4$$ My progress so far ; I worked on ...
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Prove $8abc(a+b+c)^3\leq27(a^2+bc)(b^2+ca)(c^2+ab)$

Let $a\geq0, b\geq0,c\geq0$. Prove that: $$8abc(a+b+c)^3\leq27(a^2+bc)(b^2+ca)(c^2+ab).$$ I tried to prove this inequality using only the inequality between the arithmetic mean and the geometric mean. ...
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1 answer
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Integral inequality and the Hölder inequality

Let $\mu:S\rightarrow[0, +\infty]$ be a positive measure on $S$ $\sigma$-algebra on $X$ such that $\mu(X)=1$, and let be $f,g:X\rightarrow \mathbb{R}$ be positive $S$-measurable functions such that: $$...
1 vote
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Inequalities about fourth and sixth moments of a R.V.

Which of the following inequalities are $\textbf{not always}$ true: $$(1). E|X| \geq |EX|$$ $$(2). E[X^4] \geq \frac{(EX)^4}{E[X^6]} \geq E[X]^6$$ $$(3).E[e^X] \geq E(1 + \frac{X}{1!} + \frac{X^2}{2!})...
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Applying Hölder's inequality

I have a function $a(u,v) := \int_\Omega c(x) u(x)v(x)dx$ where $\Omega \subseteq \mathbb{R}^n$ is open and bounded, $c \in C(\bar{\Omega})$, $c\geq 0$, $u,v \in L^2(\Omega)$. Now I want to show that ...
4 votes
2 answers
124 views

Confusions in Holder's Inequality

Holder's Inequality states that for nonnegative real numbers $a_1,...,a_n$ and $b_1,...,b_n$ we have $$\left(\sum_{i=1}^na_i\right)^p\left(\sum_{i=1}^nb_i\right)^q\ge \left(\sum_{i=1}\sqrt[p+q]{a_i^...
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Optimal speed for approaching red light to maximize velocity with non-uniform probability

Problem statement When I cross red lights, my goal is to being going as fast as possible when the light turns green. I am at distance $D$ from a traffic light when it turns red. Let the time length of ...
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Show that $\sum_{cyc}\frac{\sqrt{x}\left(\frac{1}{x}+y\right)}{\sqrt{1+x^2}}\ge 3\sqrt{2}$ [duplicate]

$x,y,z$ are positive reals, show that $$\sum_{cyc}\frac{\sqrt{x}\left(\frac{1}{x}+y\right)}{\sqrt{1+x^2}} \ge 3\sqrt{2} $$ Here is my approach $$\sum_{cyc}\frac{\sqrt{x}\left(\frac{1}{x}+y\right)}{\...
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1 answer
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Solving the inequality using Binomial Theorem

I had to prove the following inequality $$a^{3/5}b^{2/5}\leq 3a/5+2b/5$$ For real and non negative a and b I was able to prove it using Holder’s inequality and also AM-GM inequality. I need to know if ...
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Help with a vector inequality (Holder's / Jensen's)

Consider the non-negative vectors $c_i, a_i, b_i$, and positive constants $\alpha, \beta$ where $\alpha + \beta = 1$ Given that $c_i \leq a_i^\alpha b_i^\beta$. I can prove that $\sum_i c_i \leq \left(...
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Funny consequence of Hölder Inequality

I really like this one: From Hölder, we get that the series $\sum_{i=1}^\infty x_iy_i$ always converges for $x \in \ell^p$ and $y \in \ell^q$. Since $\ell^p \subseteq \ell^q$ whenever $p < q$, we ...
2 votes
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Proving a particular inequality using Hölder

I'm reading An Introduction to Inequalities by Beckenbach and the following inequality is left as an exercise for the reader ( Chapter 6, p. 104) : $$[|x|^n + |y|^n]^{1/n} \geq [|x|^m + |y|^m]^{1/m} $$...
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What is the tightest bound that relates the L-1 norm with the L-p norm, for $p \geq 1$ and $n\rightarrow \infty$?

I have a situation where is cheaper to compute $y = \sum_{i=1}^n x_i$ instead of $y_p = \sum_{i=1}^n x_i^p$, with $x_i \in \mathbb{R}_{>0}$ and $p\geq 1$. From this answer, using Hölder's ...
3 votes
0 answers
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An estimate using Holder's inequality

The author claims to have used Holder's inequality, but as I understand Holder's inequality is used to estimate the norm of the product of two functions yet here the result shows a sum in the ...
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2 answers
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Proving the inequality $2\mathsf E(Y^2)^2\le \mathsf E(Y^2)\mathsf E(Y)^2+\mathsf E(Y^4).$

Let $Y\in L^4$ be a non-negative real random variable. I want to prove that $$2\mathsf E(Y^2)^2\le \mathsf E(Y^2)\mathsf E(Y)^2+\mathsf E(Y^4).$$ My proof, based on the Hölder inequality, can be found ...
2 votes
1 answer
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Determine whether a function $f$ satisfying $\int_{B_{2r}\setminus B_r}|f(x)|^3dx<Cr$, for every $r$ is $L^1(B_1)$

Let $B_r=\{x\in\mathbb{R}^2:|x|<r\}$ denote the ball in $\mathbb{R}^2$ centered at the origin or radius $r$. Let $f$ and $g$ be measurable functions on $\mathbb{R}^2$ satisfying: There exists a ...
1 vote
2 answers
79 views

If $f\in L^4[0,1]$ and $\|f\|_4\leq C\|f\|_2$, then $\|f\|_2\leq C_1\|f\|_1$

I'm trying to solve the following problem. Let $\|f\|_p=\left(\int_0^1|f|^p\right)^{1/p}$ be the usual $L^p$ norm. Let $f\in L^4[0,1]$ and suppose there is a constant $C$ such that $\|f\|_4\leq C\|f\|...
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Does there exist a constant $C>0$ such that for all simple functions $f$, $\int_1^\infty|f|\leq C(\int_1^\infty|f|^p)^{1/p}$?

I'm trying to solve the following problem. Let $p\in(0,\infty)$ be fixed. Determine, with justification, whether the following statement is true or false. There exists a constant $C>0$ such that ...
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is it possible to use Holder inequality in this way?

In the book Superlinear Parabolic Problems Blow-up, Global Existence and Steady States, page 492 this equation appears in which the book says it uses Holder inequality My question is whether it's ...
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To show that $\lim_{x\to\infty}\cfrac{1}{x^{1-1/p}}\int_0^x f(t)dt=0$ via Holder's inequality. [duplicate]

I'm trying to solve the following problem but I'm stuck at one point. Let $1<p<\infty$ and $f\in L^p[0,\infty)$. Show that $$\lim_{x\to\infty}\cfrac{1}{x^{1-1/p}}\int_0^x f(t)dt=0.$$ And a hint ...
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1 answer
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How to prove $\|f*g\|_1 \leq \|f\|_1 \cdot \|g\|_1$ [duplicate]

$f*g$ denotes the convolution of $f$ and $g$. Let $f,g \in L^1(\mathbb{R}^n)$. How can I proof that $||f*g||_1 \leq ||f||_1 \cdot ||g||_1$? (Where $||\,.||_1$ is the $L^1$-norm.) I tried using the ...
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Norm with non-negativity constraint

Is it possible to solve the following problem for general norm $$ \max_{x \geq 0, \|x\| \leq \lambda} y^{\top} x $$ In the special case of infinity norm we have $$ \begin{aligned} \max_{x \geq 0,\; \...
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3 votes
1 answer
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Inequality in cyclic order : $\sum\frac{8}{(a+b)^2+4abc}+a^2+b^2+c^2\ge\sum\frac{8}{a+3}$ [closed]

Prove:$$\sum_{cyc} \frac{8}{(a+b)^2+4abc}+a^2+b^2+c^2\ge\sum_{cyc} \frac{8}{a+3}$$ where $a,~b,~c$ are positive real numbers. My thought: I think Holder's inequality will be used. But can't understand ...
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Generalization of Cauchy-Schwarz/Hölder inequality

For functions $u,v \in L^2(\Omega)$, with $\Omega \subset \mathbb{R}^n$, the Cauchy-Schwarz inequality as a special case of the Hölder inequality $(p=q=2)$ states that (1) $\Vert uv\Vert_{L^1} = \...
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123 views

Prove Holder's inequality with $0<p<r<q<\infty,\ (1-\theta)/p+\theta/q=1/r$ and $0<\theta<1 \implies \|h\|_r \leq \|h\|_p^{1-\theta} \|h\|_q^{\theta}$

I'm trying to prove Holder's inequality using that in a measure space $(X,\mu)$ for every $h:X\to \mathbb{C}$ measurable, $0<\theta<1$ and $0<p<r<q<\infty$, with $$\frac{1}{p}(1-\...
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Lower bound on the difference between the product of l1 and l2 norm

I've just occurred to a question that supposes we have two positive vectors $A$ and $B$. Then what can we say about $||A||_{l1}||B||_{l1} - ||A||_{l2}||B||_{l2}$, i.e. $\sum a_i \sum b_i - (\sum a_i^2)...
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