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Questions tagged [holder-inequality]

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

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Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $\lceil$ https://...
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Prove that $\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$ [on hold]

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$$ I tried Holder Inequality (it's only the hint to get you ...
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Inequality between two sums

As part of a research problem I am working on, I need to show the following inequality. Let $x=(x_1,\dots,x_K)$ with $x_i > 0$ for all $i$. Then, I wish to show that $$ \frac{1}{K^2}\sum_{i=1}^K ...
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Prove $\prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 $

Prove $$\begin{equation}\begin{split} \prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 \end{split}\end{equation}$$ with $a,\,b,\,c> 0$. $$\begin{equation}\begin{...
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Holder conditional inequality

we consider, on a probability space $(\Omega,\mathcal{A},P)$, two random variable $X$ and $Y$ and let $\mathcal{H} \subset \mathcal{A}$ be a $\sigma$-algebra. Let $p,q>1$ such that $\frac{1}{p}+\...
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31 views

Holder inequality for variable Lebesgue spaces

I'm reading the proof on the book: http://www.crm.cat/en/Publications/Publications/2011/Pr1093.pdf and I don't understand this sentence in page 24: Combining the above terms, and using the fact that ...
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estimating $\prod_{i=1}^k a_i\leq \max_i a_i^{p_i}$, where $\sum_i 1/p_i=1$

Let $1<p_1,\dots,p_k<\infty$ such that $\sum_{i=1}^k\frac{1}{p_i}=1$. Moreover, let $a_1,\dots,a_k\geq 0$. I have to show that $$\prod_{i=1}^k a_i\leq\max_i a_i^{p_i}$$ I want to use induction ...
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Proof with Holder's inequality

I am doing exercises about Holder and Minkowsvi inequalities and I have a doubt to an exercise that seems easy, but I am stucked. It uses Holder inequality in functions $\varphi,\psi$. Choosing $\...
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Norm of function in $L^p$ as a sup of integrals

Let $p, q$ be positive real numbers that satisfy $\frac{1}{p} + \frac{1}{q} = 1.$ I want to show that, if $\color{magenta}{\mathbf{f \in L^p}}$ $||f||_p = \sup \displaystyle \int f \cdot g$, where ...
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How to apply Holder's inequality to this problem?

I want to show the following: On a set of real numbers that has finite measure, if $a > b$ and $||f_n - f||_a \rightarrow 0$, then $||f_n - f||_b \rightarrow 0$ as well. Perhaps I should be using ...
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show this inequality to $\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$ This problem is from Iran 3rd round-2017-Algebra final exam-P3,...
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How to prove this Holder- type inequality?

I don't understand why this inequality holds (i found it in some notes): Let $ v\in L^1 \cap C_c, $ then $$ \Vert v\Vert_{L^{q}} \leq \Vert v\Vert_{L^{1}}^{\frac{1}{q}} \cdot \Vert v\Vert_{L^{\infty}}...
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A generalized Hölder inequality

I have the following execize and I would like to know if it is right: Let $u\in L^{p}(\mathbb{R^{n}}) \cap L^{q}(\mathbb{R^{n}})$ with $1\leq p \leq q\leq +\infty$. Prove that $u \in L^{r}(\mathbb{R^{...
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Prove, that for every real numbers $ x \ge y \ge z > 0 $, and $x+y+z=\frac{9}{2}, xyz=1$, the following inequality takes place

Prove, that for every real numbers $ x \ge y \ge z > 0 $, and $x+y+z=\frac{9}{2}, xyz=1$, the following inequality takes place: $$ \frac{x}{y^3(1+y^2x)} + \frac{y}{z^3(1+z^2y) } + \frac{z}{x^3(1+...
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Understanding generalized Holder inequality proof

There are questions that concerns me when I read the following proof regarding the generalized Holder inequality : Let $U$ be a subset of $\mathbb{R}$. Let $1 < p, q, r < \infty$ with $p^{-1} + ...
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Some Cauchy-Schwarz Inequalities [closed]

I am trying to learn how to deal with inequalities to prepare for a Math Olympiad and right now I am working on Cauchy-Schwarz. However, I am not that good at seeing the relationships and I don't have ...
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Hölder's inequality: What does $\|\cdot\|_1$ refer to when it's an integral?

Hölder's inequality: What does $\|\cdot\|_1$ refer to when it's an integral? In the L.H.S. of Hölder's inequality.
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Maximum value of expression $a+b+c$

If $a,b,c$ are non negative integers such that $$2(a^3+b^3+c^3)=3(a+b+c)^2.$$ Then maximum value of $a+b+c$ is ? My Try: Using Jensen Inequality Let $f(x)=x^3$. Then $f''(x)>0$ for $x>...
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Cauchy Schwarz inequality with 1 norm

Here is the argument I am making By Holder's inequality, we have for $\frac{1}{p} +\frac{1}{p^*} = 1$ $$\langle A, B\rangle \leq ||A||_p||B||_{p^*}$$ The Schatten p-norms also obey $||A||_p \geq ||...
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Exercise in Holder's inequality

The following is a problem from Royden and Fitzpatrick's Real Analysis book. Find the values of the parameter $\lambda$ for which $$ \lim\limits_{\epsilon\rightarrow0^{+}} \frac{1}{\epsilon^\...
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A lower bound for Holder's inequality and an upper bound for the reverse Holder inequality!

I know that Holder's inequality already states, for non-negative sequences ${a(n)},{b(n)}$, that $$((∑a(n)b(n))/((∑a^{p}(n))^{1/p}(∑b^{q}(n))^{1/q}))≤1$$ where $p>1$ and $(1/p)+(1/q)=1$ and ...
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$\frac{1}{3}\sum_{cyc}\frac{1}{\sqrt{1+a}}\ge\frac{1}{\sqrt{1+\sqrt[3]{abc}}}$ if $\;a, b, c\;$ are positive reals s.t. $abc\ge2^9$

$$\frac{1}{3}\sum_{cyc}\frac{1}{\sqrt{1+a}}\ge\frac{1}{\sqrt{1+\sqrt[3]{abc}}}$$ if it is given that $\;a, b, c\;$ are positive reals s.t. $abc\ge2^9$. I have tried (many) dead-end solutions. ...
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Showing that an integral operator on $L^p$ spaces has a certain norm

Let $X$ be a sigma-finite measure space, and let $k$ be a measurable function on $X\times X$. Suppose that $F(x)=\int |k(x,y)| dy$ and $G(y)=\int |k(x,y)| dx$ are in $L^\infty$. Let $1<p<\...
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Holder's Inequality with Indicator Functions probability

Suppose I have two events $A$ and $B$. If $A$ and $B$ are independent, I can say that $$\mathbb{E}[\mathbb{1}_{A}\mathbb{1}_{B}] = \mathbb{E}[\mathbb{1}_A] \mathbb{E}[\mathbb{1}_B] = Pr(A) Pr(B)$$ ...
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Show that $a^{2014}+b^{2014}\geq a^{2013}+b^{2013} $.

Let $ a, b\in \mathbb {R}_{+} $ s.t. $a^{22}+b^{22}=a^{3}+b^{3} $. Show that $a^{2014}+b^{2014}\geq a^{2013}+b^{2013} $. By Chebyshev's inequality we obtain $a^{19}+b^{19}\leq 2\Leftrightarrow b^{19}...
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Trig Integral Inequality

Show that if $f$ is Riemann integrable on $[a,b]$, then $$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2\le(b-a)\int_{a}^{b}f^2(x)\ dx.$$ I know I need to use ...
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Cauchy Schwartz Inequality Question: $(a^2+b^2)^3=c^2+d^2 \implies \frac{a^3}{c}+\frac{b^3}{d}\geq 1$

If $(a^2+b^2)^3=c^2+d^2$, prove that $\frac{a^3}{c}+\frac{b^3}{d}\geq 1$. Please help.
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Doubt in variant of Holder inequality

Let $p, q, r \in [1, \infty), r \neq \infty$ such that $1/p+1/q=1/r$. If $f \in {L(X)} ^ p$ and $g \in {L (X)}^q$. Is it true that $|f|^{p/r}<|f|^p$? According to I do not, because if I consider $...
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Use of Holder inequality in gradient estimate for harmonic function.

While reading the book "Elliptic Partial Differential Equations" by Han and Lin, I failed to understand the proof of the interior gradient estimate for harmonic functions. The theorem says that if $u$ ...
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29 views

Alternative form for Liapunov inequality

Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then $$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$ My teacher ...
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1answer
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Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Let $a,b,c\in \Bbb R^+$ such that $a+b+c=abc$. Prove that $$\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$$ Idea 1.From $a+b+c=abc\Leftrightarrow \frac{1}{ab}+...
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1answer
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Interchanging limit and integral.

Suppose $(X,\mu)$ is a probability space, $W\in L^1(X)$, $V\in L^\infty(X)$, and $V_n\to V$ in $L^2(X)$ (in my situation $V_n$ is the partial Fourier sum and so the $L^2(X)$ convergence is automatic). ...
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1answer
41 views

A Cauchy-Schwarz-type inequality for $\int\prod_n|f_n|$

If $X_1,X_2$ have finite second moments then Cauchy-Schwarz gives $\langle |X_1||X_2|\rangle^2 \leq \langle |X_1|^2\rangle \langle |X_2|^2\rangle $ If $(X_n)_{n=1}^N$ have their $N$th moments is it ...
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Inequality with $(x+y)(y+z)(z+w)(w+x)=1$

Let $x,y,z,w>0$ and such that $$(x+y)(y+z)(z+w)(w+x)=1.$$ Show that $$\sqrt[3]{xyz}+\sqrt[3]{yzw}+\sqrt[3]{zwx}+\sqrt[3]{wxy}\le 2.$$ I'm trying to use Holder's inequality $$(\sqrt[3]{xyz}+\sqrt[3]...
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Prove $\int_Rfg\,dm\leq\|f\|_p^{1-p/r}\|g\|_p^{1-q/r}(\int_Rf^pg^q\,dm)^{1/r}$, where $1\leq p\leq\infty$ and $\frac1{r}=\frac1{p}+\frac1{q}-1$

Let $f$, $g$ be positive real functions. And $f \in L^p(R)$, $g \in L^q(R)$, and $1 \leqslant p,q <\infty$. Then $fg \in L^1(R)$ and $$ \int_R fg \,dm\;\leqslant\; \|f\|_p^{1-p/r}\|g\|_p^{1-q/r}\...
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How do I show this $xy\leq \frac{1}{p}x^p+\frac{1}{q}y^q.$ without using concavity of log function?

let reals $p, q >1$ and $x, y$ are positive real numbers with $1/p+1/q=1$ I want to show $xy\leq \frac{1}{p}x^p+\frac{1}{q}y^q.$ without using concavity of $\log$ as shown below Proof with ...
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Suppose that $p>1$, and ${a(n)},{b(n)},{ω(n)}$ are nonnegative sequences.

if $∑_{n=1}^{N-1}a(n)=∑_{n=1}^{N-1}b(n)$. Is there a relation between $∑_{n=1}^{N-1}ω^{1-p}(n)a^{p}(n)$ and $∑_{n=1}^{N-1}ω^{1-p}(n)b^{p}(n)$. p.s. I tried obtaining an inequality similar reverse ...
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1answer
35 views

The Convergence of Convolution of $f$ and $g$

I am dealing with a question requiring me to prove that if $f,g\in L_{2}$, then the convolution is defined everywhere, bounded, and continuous. Moreover, it will converge to $0$ as $|x|\rightarrow\...
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1answer
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Let $T(f)(y)=\int_0^\infty K(x,y)\cdot f(x)dx$, $\,$ show $\,$ $\Vert T(f) \Vert _p \le C\cdot \Vert f \Vert _p$

Now here is the full statement. Let $K:(0, +\infty) \times (0, +\infty) \rightarrow \Bbb R$ be a Lebesgue measurable function with $K(kx,ky)=k^{-1}K(x,y)$ for every $k>0$ and let $$\int_0^\...
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A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$? [duplicate]

Let $a_1,\dots,a_d,b_1,\dots,b_d$ be positive real numbers. Then $$ \prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$$ and equality holds if and only if $\frac{a_i}{a_j}=\frac{...
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1answer
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Show that the functional is continuous

Task: Check if the functional F: $$F(f) = \int_{-1}^1 f(t) sgn(t) dt.$$is continuous on the space $\mathbb E=L_2(-1, 1) $: Solution: I need to find $M>0$ and show the inequality: $$\vert F(f) \...
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2answers
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Find the minimum value of $\sqrt {2x^2+2y^2} +\sqrt {y^2+x^2-4y+4} +\sqrt {x^2+y^2-4x-4y+8}$

Given that $0\lt x\lt 2$ and $0\lt y\lt 2$ then find the minimum value of $$\sqrt {2x^2+2y^2} +\sqrt {y^2+x^2-4y+4} +\sqrt {x^2+y^2-4x-4y+8}$$ My try: On factorisation we need minimum value of $$\...
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1answer
37 views

Hint: Hölder Inequality with $\theta$ application

Let $\frac{1}{r}=\frac{\theta}{p}+\frac{1-\theta}{q}$ where $1\leq p<r<q\leq \infty$ and $\theta \in ]0,1[$, as well as $f \in L^{p} \cap L^{q}$. Show that $||f||_{r}\leq||f||_{p}^{\theta}\times|...
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1answer
25 views

Showing $f:=f_{1},…,f_{n}\in L^{p}(\mu)$ for $f_{i} \in L^{p_{i}}$

Let $f_{1},...,f_{n}:X \to \bar{\mathbb R}$ measurable, while $f:=f_{1}\times...\times f_{n}$, and $p_{1},...,p_{n} \in [1,\infty]$ where $f_{i}\in L^{p_{i}}(\mu), \forall i\in \{1,...,n\}$. Note $\...
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1answer
30 views

An application of Holder's Inequality

Suppose $1\leq p,q\leq \infty$ and $1/p+1/q=1$. Let $f\in\mathcal{L}^p(E)$. Show that $f=0$ a.e. if and only if \begin{align*} \int_E f\cdot gdm=0 \end{align*} for all $g\in \mathcal{L}^q(E)$. ...
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3answers
290 views

Prove that $({a\over a+b})^3+({b\over b+c})^3+ ({c\over c+a})^3\geq {3\over 8}$

Let $a,b,c$ be positive real numbers. Prove that $$\Big({a\over a+b}\Big)^3+\Big({b\over b+c}\Big)^3+ \Big({c\over c+a}\Big)^3\geq {3\over 8}$$ If we put $x=b/a$, $y= c/b$ and $z=a/c$ we get $xyz=1$ ...
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1answer
92 views

$\frac{a}{\sqrt{a^2 + 8bc}} +\frac{b}{\sqrt{b^2 + 8ac}} + \frac{c}{\sqrt{c^2 + 8ab}} \ge \frac{1}{\sqrt{a^3+b^3+c^3 + 24abc}}$ is true?

In one of the solutions of a problem in this site: https://artofproblemsolving.com/wiki/index.php?title=2001_IMO_Problems/Problem_2 It is used the following: If $a,b,c$ are positive real numbers ...
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1answer
40 views

Showing a product of two Lebesgue integrals is $\geq 1$ if the product of the integrands is $\geq 1$

Let $\mu$ be a probability measure on a set $X$, i.e. $\mu(X)=1$, and let $f$ and $g$ be positive measurable functions on $X$. Show that if $fg\geq1$, then the integral of $f$ times the integral of $...
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1answer
47 views

Expected value Holder inequality

I have the following question; how do you prove this statement? I know that it is probably something related to the Holder inequality, but I couldn't figure out how to use it in this case. Let $p,q &...
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1answer
42 views

$L_p$ Norms and Holder's Inequality question

Suppose that $-∞ < a < b < ∞$ and $1 < p < q < ∞$. Let $$L_p[a,b] = \{ f :\Bbb R \to \Bbb R : \left( \int_a^b\left|f\left(x\right)\right|^p~\mathrm dx\right)^{\frac{1}{p}} < ∞ \}....