Questions tagged [holder-inequality]
Proving or manipulations with inequalities by using Hölder's inequality.
287
questions
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Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q $ for any $x \in \mathbb R^{d}$
Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q $ for any $x \in \mathbb R^{d}$
How do you prove this using Holder's inequality?
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0answers
43 views
Using Holder and C-S to prove the radical inequality.
For example$:$
For $x,y,z>0$ and $xy+yz+zx\geqq 3.$ Prove that$:$
$$\frac{x}{\sqrt{4x + 5y}} + \frac{y}{\sqrt{4y + 5z}} + \frac{z}{\sqrt{4z + 5x}} \geqq 1$$
It is easy and you can see my two ...
1
vote
1answer
37 views
Holder's Inequality for $0<p<1$ [closed]
I would like to ask about the constant $C$ for which the following inequality holds.
Let $0<p<1$ and let $q$ be its Hölder conjugate, that is $1/p+1/q=1$, then
Is there $C>0$ such that
$$||...
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0answers
23 views
Prove that $\frac{3}{50}<\int_{0}^{1}\exp\Big(-\operatorname{W^2(x)}\Big)\operatorname{W^e(x)}dx$
At the beginning I was thinking to the Laplace transform of the Lambert's function as there is no easy to way express this I propose this similar problem :
$$\frac{3}{50}<\int_{0}^{1}\exp\Big(-\...
3
votes
3answers
73 views
prove that $3(a+b+c) \geq 8(a b c)^{1 / 3}+\left(\frac{a^{3}+b^{3}+c^{3}}{3}\right)^{1 / 3}$
Question -
Suppose a,b,c are positive real numbers , prove that
$3(a+b+c) \geq 8(a b c)^{1 / 3}+\left(\frac{a^{3}+b^{3}+c^{3}}{3}\right)^{1 / 3}$
(Thailand $2006$)
My attempt -
we can assume ...
2
votes
2answers
62 views
Olympiad Inequality: Cauchy Schwartz
Can someone please help me with some hint or the solution of this problem below:
For $a,b,c,k >0$
$ \frac{a}{\sqrt{ka+b}} +\frac{b}{\sqrt{kb+c}} + \frac{c}{\sqrt{kc+a}} < \sqrt{\frac{(k+1)}{k}(...
2
votes
1answer
92 views
$\sum\limits_{cyc}\frac{a}{\sqrt{a+3b}}\geq\frac{a+b+c}{2}$ for $a+b+c+abc=4$
Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc\neq0$ and $a+b+c+abc=4.$ Prove that:
$$\frac{a}{\sqrt{a+3b}}+\frac{b}{\sqrt{b+3c}}+\frac{c}{\sqrt{c+3a}}\geq\frac{a+b+c}{2}.$$
The ...
3
votes
1answer
104 views
If $a+b+c+d=4$ Prove that $ \sqrt{\frac{a+1}{a b+1}}+\sqrt{\frac{b+1}{b c+1}}+\sqrt{\frac{c+1}{c d+1}}+\sqrt{\frac{d+1}{d a+1}} \geq 4 $
Question -
Let $a, b, c, d$ be non-negative real numbers with sum 4. Prove that
$
\sqrt{\frac{a+1}{a b+1}}+\sqrt{\frac{b+1}{b c+1}}+\sqrt{\frac{c+1}{c d+1}}+\sqrt{\frac{d+1}{d a+1}} \geq 4
$
My work ...
1
vote
1answer
60 views
Olympiad inequalities. [duplicate]
I am trying to solve this problem but unable to. Can someone please give some hint or help.
I have to use holder inequality.
For $a,b,c$ positive real numbers prove.
$ \frac{a^6}{b^2+c^2} + \...
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1answer
35 views
Clarifying a step in the equality case of the Hölders inequalite
Can anyone please explain to me why $f'g' = \frac{1}{p}f'^p + \frac{1}{q}g'^q$ iff $f'^p = g'^q$ in this post On the equality case of the Hölder and Minkowski inequalites
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0answers
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Reciprocal of Hölder's inequality [duplicate]
Let $(X,S,\mu)$ be a $\sigma$-finite measure space, and $g \in M(X,S)$ such that $gs \in L_1(\mu)$ for any simple function $s \in L_p(\mu)$, with $p \in (1, \infty)$. Suppose $A \geq 0$ exists such ...
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0answers
49 views
Applying Hölder inequality to prove $\Vert x \Vert_p \leq n^{\frac{1}{p}-\frac{1}{q}} \Vert x \Vert_q$ [duplicate]
As mentioned in the title I have to prove the inequality: $$\Vert x \Vert_p \leq n^{\frac{1}{p}-\frac{1}{q}} \Vert x \Vert_q.$$ The expressions $\Vert \cdot \Vert_p$ and $\Vert \cdot \Vert_q$ are the ...
1
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1answer
61 views
Reciprocal of Holder's inequality
Does anyone know how to do this exercise? I couldn't do it.
Let $(X,S,\mu)$ be a $\sigma$ -finite measure space and $g\in M(X,S)$ such that $gs\in\mathcal{L}_1$ for any simple function $s\in\mathcal{...
3
votes
4answers
87 views
An inequality involving homogeneous polynomials
Let $x_1, x_2, \dots x_k \ge 0$ be non-negative real numbers.
Does it follow that
$$k \left( \sum_{i=1}^k x_i^3 \right)^2 \ge \left( \sum_{i=1}^k x_i^2 \right)^3 ? $$
This seems like something that ...
3
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4answers
90 views
Roots of the equation $(1-4x)^4+32x^4=\frac{1}{27}$
find all real roots of the equation $(1-4x)^4+32x^4=\dfrac{1}{27}$
i try to use binomial expansion and am-gm inequality but i don't know how to do next.
2
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1answer
105 views
prove that $\frac{2}{b(a+b)}+\frac{2}{c(b+c)}+\frac{2}{a(c+a)} \ge \frac{27}{(a+b+c)^2}$
Prove that $$\frac{2}{b(a+b)}+\frac{2}{c(b+c)}+\frac{2}{a(c+a)} \ge \frac{27}{(a+b+c)^2},$$ where $a,b,c$ are positive reals.
After applying AM-GM I got
$$
\frac{2}{b(a+b)}+\frac{2}{c(b+c)}+\frac{2}{...
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votes
1answer
76 views
If $a^2+b^2+c^2 =3$, find the minimum value of $a+b+c$
Given that their are 3 positive real numbers $a,b,c$, such that $a^2+b^2+c^2=3$, find the minimum value of $a+b+c$
8
votes
3answers
131 views
For non-negative reals $a$, $b$, $c$, show that $3(1-a+a^2)(1-b+b^2)(1-c+c^2)\ge(1+abc+a^2b^2c^2)$
A 11th grade inequality problem:
Let $a,b,c$ be non-negative real numbers. Prove that
$$3(1-a+a^2)(1-b+b^2)(1-c+c^2)\ge(1+abc+a^2b^2c^2)$$
Do you have any hints to solve this inequality? Any ...
1
vote
2answers
73 views
Inequality with condition $2(x^2+y^2+z^2)\leq 3(x+y+z-1)$
If $x,y,z>0$ such that $2(x^2+y^2+z^2)\leq 3(x+y+z-1)$, then find the minimum value of:
$$S=(x+y+z)\bigg(\frac{1}{\sqrt{2x^3+x}}+\frac{1}{\sqrt{2y^3+y}}+\frac{1}{\sqrt{2z^3+z}}\bigg)$$
What I ...
1
vote
1answer
35 views
Find an upper bound on the expectation of squared norm given an upper bound on the expectation of norm
For non-independent random vectors $X, Y$, I have an upper bound on the expectations $E[\|X\|_2] \leq a, E[\|Y\|^2_2] \leq b$. How can I compute an upper bound for $E[\|X^\intercal Y\|_2]$ or $E[\|X^\...
4
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2answers
105 views
Prove $\sqrt[3]{\frac{(a^4+b^4)(a^4+c^4)(b^4+c^4)}{abc}}\ge \sqrt{\frac{(a^3+bc)(b^3+ca)(c^3+ab)}{1+abc}}$
If $a,b,c>0$, prove that:
$$\sqrt[3]{\frac{(a^4+b^4)(a^4+c^4)(b^4+c^4)}{abc}}\ge \sqrt{\frac{(a^3+bc)(b^3+ca)(c^3+ab)}{1+abc}}$$
My try: I used $a^4+b^4 \geq ab(a^2+b^2)$ to get:
$$(a^4+b^4)(a^4+...
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votes
0answers
30 views
Estimates with $L^n$ norm
this is my question:
Show that $\epsilon ||f||_{L^n(\Omega')}\leq \epsilon^{\frac{x}{n}}$, where $x$ is a positive number, using these two facts:
$\int_{\Omega}|f|^x\leq C $
$|f|\le\dfrac{C}{\...
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1answer
26 views
Holder inequality Proof Explanation
I don't understand why
$$ptf(x)h(x)^{p-1}\leq (h(x)+tf(x))^p-h(x)^p$$.
3
votes
2answers
395 views
Barnard and Child inequality exercise
Prove that,
$$3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)≥abc(a+b+c)^3$$
For positive $a,b,c$
The exercises in this book are making me crazy.
Any help would be appreciated.
My attempts:
I opened the LHS ...
0
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1answer
42 views
Type of Holder Inequality
I am struggling to find a proof of the following inequality which is a modification of the generalized Holder inequality
Let $(\Omega, \mathcal F, \mu)$ a mesurable space. Let $ \alpha \in [0,1]$ ...
0
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1answer
27 views
How to prove $\sum_{i=1}^{k}x_i^2\ge k$ if $\sum_{i=1}^{k}\frac{1}{x_i}=k$ and $\min (x_i-1)^2$ is sufficiently small?
How to prove $\sum_{i=1}^{k}x_i^2\ge k$ if $\sum_{i=1}^{k}\frac{1}{x_i}=k$ and $\min (x_i-1)^2$ is sufficiently small?
In $k=2$ case it is true.
$\frac{1}{x}+\frac{1}{y}=2$ implies $y = \frac{x}{2x-...
0
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1answer
31 views
Hölder' inequality for random variables
$X_n$ and $X$ is integrable random variable, for some $p>1$, we have
$|X_n-X|^p\leq 2^{p-1}(|X_n|^p+|X|^p)$.
Does anyone know how to derive this inequality? The solution tells me it's Hölder's ...
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1answer
42 views
Solving an Inequality problem [closed]
I have these two problems among other similar question that I haven't figured out how to solve. I believe I could use the AM–GM inequality to solve this but I'm unsure how that would look.
Let $a$ ...
1
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0answers
13 views
Norm of adjoint via Hölder conjugates
Let $T$ be a linear operator between finite-dimensional complex vector spaces $A, B$. Then it is mentioned on Wikipedia that for any $(p, q)$ there is a relation between the operator norms
$$||T : \...
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0answers
15 views
Wanted: an inequality with variable exponent
[Wanted] I want to estimate over the following quantity
$ | a + b | ^ {2-2p (x)}$
Where:
• $a$ and $b$ are vectors in $\mathbb{R} ^{n}$ (!?)
• $p$ is a function $C^{1}$ in a limited domain in $\...
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0answers
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$||f||_r^{\lambda r} ||f||_s^{(1-\lambda)s}\le(\max\{||f||_r,||f||_s\})^p$ inequality for $r<p<s$ and $p=\lambda r+(1-\lambda)s$
Let $r<p<s$ and $p=\lambda r+(1-\lambda)s$ for some $\lambda \in(0,1)$
We denote $||f||_{L^p}$ as $||f||_p$.
Why do we have $||f||_r^{\lambda r} ||f||_s^{(1-\lambda)s}\le(\max\{||f||_r,||f||_s\...
2
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3answers
173 views
Cyclic Inequality $(x^8+1)(x^4+1)(y^8+1)(y^4+1)(z^8+1)(z^4+1)≥((x^2+1)(y^2+1)(z^2+1))^2$
I have been challenged to prove the following:
If $xy+yz+zx=3$ for positive $x,y,z$, then $(x^8+1)(x^4+1)(y^8+1)(y^4+1)(z^8+1)(z^4+1)≥((x^2+1)(y^2+1)(z^2+1))^2$.
Initial attempts at AM-GM are not ...
1
vote
3answers
91 views
Knowing that $\prod_{i = 1}^na_i = 1$, prove that $\prod_{i = 1}^n(a_i + 1)^{i + 1} > (n + 1)^{n + 1}$.
Given natural $n$ $(n \ge 3)$ and positives $a_1, a_2, \cdots, a_{n - 1}, a_n$ such that $\displaystyle \prod_{i = 1}^na_i = 1$, prove that $$\large \prod_{i = 1}^n(a_i + 1)^{i + 1} > (n + 1)^{n + ...
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0answers
43 views
Holder semi norm estimate for a solution of Poisson's equation in the half space
Let $u\in C^{2,\alpha}_0(\overline{\mathbb{R}^n_+})$, $0<\alpha<1$, such that
$$-\Delta u=f \quad\text{in }\quad \overline{\mathbb{R}^n_+}$$
$$ u=0 \quad\text{in }\quad \partial\overline{\...
1
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1answer
62 views
Holder inequality with $q = \infty$
Assume we have a probability space $(\Omega, \mathscr{F},P)$. As a part of a proof I found the following:
If $X \in L^1$, then for any $\epsilon >0$, there exists some $\delta
> 0$ such that ...
0
votes
1answer
38 views
How to prove this inequality with exponents?
Let $G = |Q_1+Q_2|^{p-1}(Q_1+Q_2) - |Q_1|^{p-1}Q_1-|Q_2|^{p-1}Q_2$
where $Q_1,Q_2:\mathbb{R}^d\to \mathbb{R}$ are positive valued functions and $p>2.$
I want to show that,
$$|G| \leq K\left( |Q_1|^{...
1
vote
1answer
139 views
Show that for $p>1,$ $\mathbb{E}[Y^p] \leq \big(\frac{p}{p-1}\big)^p\mathbb{E}[X^p]$
Let $X$ and $Y$ be non-negative r.v. defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that
$$\mathbb{P}(Y>y)\leq\frac1y \int_{Y\geq y}^{} X d\mathbb{P} $$
Show that ...
1
vote
1answer
33 views
if $\{f_n\}\to f$ in $L^{p_2}(E)$ then $\{f_n\}\to f$ in $L^{p_1}(E)$.
Assume that $E$ has finite measure and $1 \le p_1 < p_2 \le \infty$. Show that if $\{f_n\}\to f$ in $L^{p_2}(E)$ then $\{f_n\}\to f$ in $L^{p_1}(E)$.
my attempt: let $p=\frac{p_2}{p_1}$ and $1=\...
0
votes
1answer
31 views
show that $\|f\|_p = \max_{g\in L^q(E)\\\|g\|_q\le 1} \int_E f.g~d\mu$
For $1\le p < \infty$, $q$ conjugate of $p$ and $f\in L^p(E)$ show that;
$$\|f\|_p = \max_{g\in L^q(E)\\\|g\|_q\le 1} \int_E f.g~d\mu$$
My attempt:
(Do I need anything to assume about $E$ to be ...
0
votes
2answers
65 views
Prove inequality by inequality of means
For $ a, b $ and $ c $ positive reals, prove that:
$$\frac{a+ \sqrt{ab}+\sqrt[3]{abc}}{3} \leq \sqrt[3]{a\left(\frac{a+b}{2}\right)\left(\frac{a+b+c}{3}\right)}$$
Solution: By AM-GM and Hölder
$RHS=\...
0
votes
1answer
72 views
$a, b, c$ are three reals such that $ab + bc + ca = 3abc$. Prove that $\sum_{cyc}\sqrt{2(a + b)} \ge \sum_{cyc}\sqrt{\frac{a^2 + b^2}{a + b}} + 3$.
$a$, $b$, $c$ are three reals such that $ab + bc + ca = 3abc (a + b, b + c, c + a > 0)$. Prove that $$\large \sum_{cyc}\sqrt{2(a + b)} \ge \sum_{cyc}\sqrt{\frac{a^2 + b^2}{a + b}} + 3$$
Let $$ax = ...
0
votes
1answer
59 views
Inequality $\sum_{cyc} \sqrt{\frac{a}{b+c+d}}>2$
Let $a,b,c,d>0$. I want to prove that $$\sum_{cyc} \sqrt{\frac{a}{b+c+d}}>2.$$
I try using Hölder's inequality : $$\left(\sum_{cyc} b+c+d\right)\left(\sum_{cyc} \sqrt{\frac{a}{b+c+d}}\right)^2\...
0
votes
1answer
39 views
if $\mu(X)=1$ , $f\in L^1$, then $f\in L^\infty$?
if $\mu(X)=1$ , $f\in L^1$, then is true that $f\in L^\infty$?
Basically, I was wondering if I can write
$$\int_X f d\mu \le \mu(X). \|f\|_\infty < \infty$$
or can we claim $\|f\|_\infty =1$? I'm ...
0
votes
1answer
33 views
Hölder inequality with Lebesgue measure
Let $(\Omega, \mathcal A, \mu)$ be a measure space.
Let $p, q, r \in [1, \infty]$ with $1/p + 1/q + 1/r = 2$. Let $f \in L^p(\mathbb R^n), g \in L^q(\mathbb R^n), h \in L^r(\mathbb R^n)$.
Why is ...
7
votes
8answers
186 views
Prove that the minimum values of $x^2+y^2+z^2$ is $27$ with given condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.
Question: Prove that the minimum values of $x^2+y^2+z^2$ is $27$, where $x,y,z$ are positive real variables satisfying the condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.
From AM$\ge$ GM, we ...
2
votes
1answer
84 views
Proving an interesting inequality with square roots
Let $a,b,c>0$ be real numbers such that $c \geq a \geq b$ and $a^2 \geq bc$. Show that
$$\frac{\sqrt{a^2b+b^2c}}{a+c}+\frac{\sqrt{b^2c+c^2a}}{b+a}+\frac{\sqrt{c^2a+a^2b}}{c+b} \geq \frac{\sqrt a ...
2
votes
3answers
295 views
Prove that $(x + \sqrt[3]{abc})^3 \le (x + a)(x + b)(x + c) \le ( x + \frac{a + b + c}{3})^3$
Let $x,$ $a,$ $b,$ $c$ be nonnegative real numbers. Prove that
$$(x + \sqrt[3]{abc})^3 \le (x + a)(x + b)(x + c) \le \left( x + \frac{a + b + c}{3} \right)^3.$$
I know that this problem is a RHS-AM-...
0
votes
1answer
15 views
If $\lim \int_0^1 |f_n(x)|=0$ and $|f_n|^2\leq g$ where g is integrable on [0,1], then $\lim \int_0^1 |f_n|^2=0$
If $\lim \int_0^1 |f_n(x)|=0$ and $|f_n|^2\leq g$ where g is integrable, then $\lim \int_0^1 |f_n|^2=0$
Does anyone have any hints on how to begin?
I think we have to use Hölder's or Dominated ...
0
votes
1answer
69 views
Does Holder's inequality require finite measure space?
I read that when the domain of functions is compact, we have the following:
$$
C[a,b] \subseteq L^{\infty}[a,b] \subseteq L^2[a,b] \subseteq L^1[a,b]
$$
and which uses Holder's inequality for the ...
4
votes
1answer
123 views
Understanding proof when equality holds in Minkowski's inequality
I have a simple question about a fact that is constantly mentioned when we have $\|f+g\|_p = \|f\|_p + \|g\|_p$ in $L^p(\mu)$ space for $1<p<\infty$ (I hope someone find this useful).
I read ...
