Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [holder-inequality]

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

0
votes
1answer
20 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 $...
0
votes
1answer
27 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 &...
-1
votes
1answer
19 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}} < ∞ \}....
7
votes
2answers
94 views

Proving $\sum_{\text{cyc}} \frac{a}{b^2+c^2+d^2} \geq \frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$

Prove that $$\frac{a}{b^2+c^2+d^2}+\frac{b}{a^2+c^2+d^2}+\frac{c}{a^2+b^2+d^2}+\frac{d}{a^2+b^2+c^2}≥\frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$$ What I tried, was to say that $a^2+b^2+c^...
1
vote
0answers
28 views

Holder inequality is equality for $p =1$ and $q=\infty$

Suppose $p=1$ and $q=\infty$, and the right hand side of Holder inequality is finite. Then, Holder inequality is equality iff $|g| = ||g||_\infty$ a.e. on $\{x: f(x) \not=0\}$. And here is the ...
1
vote
1answer
26 views

How can I solve this inequality using convexity?

Given $a, b, c\ge 0$ and $x, y, z> 0$ and $a + b + c = x + y + z$. Show that $$a ^ 3 / x ^ 2 + b ^ 3 / y ^ 2 + c ^ 3 / z ^ 2 \ge a + b + c$$ prove inequality using convexity
1
vote
1answer
23 views

A proof of Hölder's inequality and trying to understand this

Let's state the Hölder's inequality in the following way: \begin{equation*} |\sum_{k=1}^n x_k y_k| \leq (\sum_{k=1}^n |x_k|^p)^{1/p} (\sum_{k=1}^n |y_k|^q)^{1/q} \end{equation*} where $1\leq p < \...
0
votes
3answers
84 views

Prove that if $x_1 + x_2 + … + x_n = n$, then $x_1^k + x_2^k + … + x_n^k \ge n$

$x_1, x_2, ..., x_n \in \mathbb R$ are nonegative and $k \in \mathbb R$, $k \ge 1$. Prove that if $x_1 + x_2 + ... + x_n = n$, then $x_1^k + x_2^k + ... + x_n^k \ge n$. I tried to find the smallest ...
1
vote
1answer
31 views

Reverse Holder Inequality $\|fg\|_1\geq\| f\|_{\frac{1}{p}}\|g\|_{-\frac{1}{p-1}}$

Let $p\in(1,\infty)$ and $(X,\mathcal{F},\mu)$ a measure space such that $\mu(X)\not=0$. Let $f,g:X\to\mathbb{R}$ be such that $g\not=0$ a.e., $\|fg\|_1<\infty$ and $\|g\|_{-\frac{1}{p-1}}<\...
0
votes
2answers
49 views

problem regarding application of Jensen's inequality

question: For $a,b,c,d \in \mathbb{R^+}$ with $a+b+c+d = 4$, Prove $\displaystyle \sum\dfrac{a}{b(b+1)}\geq \dfrac{8}{(a+c)(b+d)}$ my attempt: $f(x)= \dfrac{1}{x^2+x}=\dfrac{1}{x(x+1)}$ is convex ...
0
votes
1answer
115 views

Prove the inequality $\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$

Let $a;b;c\in R^+$ such that $ab+bc+ca>0$. Prove that $$\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$$ I have seen the similar question is $$\frac{a^2+1}{b+c}+\...
2
votes
2answers
251 views

Prove the inequality using Chebyshev's Inequality

If $a,b,c \in(0,\infty)$, then prove that: $$9(a^3+b^3+c^3)\ge(a+b+c)^3$$ I was trying to prove this inequality using Chebyshev's Inequality and assuming $a\ge b \ge c$ but to no avail. Can please ...
0
votes
1answer
31 views

Calculating Holder's Inequality for Sums with Exponents

I'm a little confused about the procedure for calculating Holder's Inequalities for Sums with Exponents. For example, I tried to apply Holder's Inequality as follows $$(\sum_{j=1}^{T}p_{j}^{(1/q) + (...
0
votes
1answer
10 views

Let F be an absolutely continuous function on [0,1] and $F’\in L^p([0,1])$

Let F be an absolutely continuous function on [0,1] and $F’\in L^p([0,1])$ For some $p\in (1,\infty)$ relative to Lebesgue measure. show that: $\lim_{x\rightarrow0^+}(F(x)-F(0))x^{-1/q}=0$. Given $\...
0
votes
1answer
26 views

No closest point to the subspace in $L_1([0,1])$

Define $S_1=\{f\in L_1([0,1]),\int_{0}^1 xf(x) dx=0 \}$ . I want to show for every $\epsilon>0$, there exist $f$ in $S_1$ with $\lVert f-1\rVert_1 \leq1/2+\epsilon$, but there's no $f$ with $\...
3
votes
4answers
116 views

Let $f\colon\Bbb{R}^2\to \Bbb{R}$ such that $|f(x)-f(y)|\leq \Vert x-y\Vert^2.$ Prove that $f$ is a constant

Edit: Several questions of this type have been asked here before but not on the same domain $\Bbb{R}^2.$ Please, how do I deal with a function of this type or could anyone show me a reference or a ...
0
votes
3answers
38 views

Proving $f \in L^{2}[0, 1] = 0$ a.e. if integral of $x^{n}f(x)$ is 0 for each n

Let $f \in L^{2}[0, 1]$ satisfy $\int_{0}^{1} x^{n} f(x) dx = 0$ for each $n = 0, 1, 2, ...$. Show that $f = 0$ a.e. I know this will involve Holder's Inequality somewhere. So far it's clear that ...
3
votes
1answer
20 views

How to prove positive semi-definiteness or integral inequality

I am trying to show that the following matrix is positive semi-definite (PSD): $$H=\left[\begin{array}{cc} \int v(x)\,dx & \int xv(x)\,dx\\ \int xv(x)\,dx & \int x^2v(x)\,dx \end{array}\right]...
2
votes
2answers
165 views

Prove that: $xy\sqrt{z}+yz\sqrt{x}+zx\sqrt{y}\geq x+y+z$

Let $x$,$y$ and $z$ are positive and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\leq 3$$ Prove that: $$xy\sqrt{z}+yz\sqrt{x}+zx\sqrt{y}\geq x+y+z$$ The things I have done so far $$3\geq \sum \limits_{cyc}\...
0
votes
2answers
59 views

If $a$, $b$ and $c$ are positive then $\sum\limits_{cyc} \frac{a^{2}}{b^{2}+c^{2}+bc}\geq 1$

If $a$, $b$ and $c$ are positive then $\sum\limits_{cyc} \frac{a^{2}}{b^{2}+c^{2}+bc}\geq 1$. I tried to solve this problem by C-S. But I can't sovle it. Things I have done so far: $\sum\limits_{cyc}...
0
votes
1answer
33 views

Showing an inequality in the Banach space $C^1([a,b])$?

Given $f \in C^1([a,b]),$ how can it be shown that $$\|f\|_\infty^2 \le \dfrac{\|f\|_2^2}{b-a} + 2 \|f\|_2 \|f'\|_2?$$ I suppose I could use Cauchy-Schwarz, but how to handle the derivative? And I ...
0
votes
2answers
49 views

Equivalence between Hölder norms

Let us consider the following two norms: $$ \left\lVert f\right\rVert_\alpha = \left\lVert f\right\rVert_\infty + \displaystyle{\sup_{\substack{x,y \in U \\ x \neq y}} \frac{\left| f(x) - f(y))\right|...
-1
votes
1answer
55 views

Counterexamples to the inequality $\int_\Omega |f(x)|^p d\mu(x) \leq \left(\int_\Omega |f(x)| d\mu(x)\right)^p$ with $p\geq1$ [closed]

Let $(\mu,\Sigma,\Omega)$ be a measure space. Since we have the inequality $\sqrt{\frac{a^2+b^2}{2}} \geq \frac{a+b}{2}$, one may natually think that the following is true: $$ \int_\Omega |f(x)|^p d\...
1
vote
1answer
126 views

Prove that $\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$.

If $x,y,z \in[0,1/2]$, with $x+y+z=1$, then prove that: $$\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$$ OK so... I've tried to square the expression ...
1
vote
1answer
48 views

Prove generalised Hölder's inequality without calculus or analysis.

It is the generalised Hölder's inequality.I saw many analytical proofs in this site but I don't know analysis. So I need a basic proof. I proved it by A.M.-G.M. for $m=n=3$. Please help me. Proof when ...
4
votes
1answer
51 views

Algebraic significance of Holder conjugates

Consider Holder conjugate exponents $p$ and $q$, i.e., $\frac{1}{p} + \frac{1}{q} = 1$. Multiplying through by $pq$ gives $p + q = pq$. So conjugate exponenets are just those real numbers whose ...
0
votes
0answers
22 views

Applying Holder's inequality

Is it possible to apply the Holder's inequality to prove that \begin{align} ax^3+C(a)x\leq2ax^3+\tilde{C}(a) \end{align} (where $C(a),\tilde{C}(a)$ are some constants depending only on $a$, and they ...
1
vote
1answer
38 views

How to show in $R^2$, the $f(x,y)=(x^4+y^4)^{1/4}$ defines a norm on $R^2$. [duplicate]

How to show in $R^2$, the $f(x,y)=(x^4+y^4)^{1/4}$ defines a norm on $R^2$, i.e., how to prove the inequality: $$((x_1+x_2)^4+(y_1+y_2)^4)^{1/4}\le (x_1^4+y_1^4)^{1/4}+(x_2^4+y_2^4)^{1/4}$$
4
votes
1answer
281 views

When does the equality hold in the Holder inequality?

I am considering the series case. In the Holder inequality, we have $$\sum|x_iy_i|\leq\left(\sum|x_i|^p\right)^{\frac1p} \left(\sum|y_i|^q\right)^{\frac1q},$$ where $\frac1p+\frac1q=1,~p, q>1$. ...
0
votes
1answer
63 views

Generalized Hölder inequality: Application

how can I prove the following inequality (we can use Hölder inequality) ? With quantities all positive and $n\geq 1$ we have $$ (a+b)^{n-1}(A+B) \geq ((a^{n-1}A)^{1/n} + (b^{n-1}B)^{1/n})^n $$ ...
0
votes
1answer
64 views

Relation between convolution and $L^p$ norms

I have to prove that for any $g \in L^1(\mathbb{R})$ and for any function $u(t)$ bounded and continuous, the following inequality holds: $$sup_{t \in \mathbb{R}}|(g \ast u)(t)| \leqslant ||g||_1\cdot|...
-1
votes
1answer
48 views

Corollary of Holder inequality

I don't understand how applicate H-inequality to prove this thing: $|(\rho_n\star f)(x)-f(x)|\leq\int|f(x-y)-f(x)|\rho_n(y)dy\leq\biggl(\int|f(x-y)-f(y)|^p\rho(y)dy\biggr)^{1/p},$ where $\rho_n$ is a ...
2
votes
1answer
40 views

There exist constants such that $\|v\|_{0,2}≤c_1\|v\|_{1,2}+c_2 |u|$

How can I show that there exist constants $c_1,c_2 >0$ such that for all $v \in H^1(a,b)$ the inequality $$\|v\|_{0,2}≤c_1\|v\|_{1,2}+c_2 |u|$$ holds where $u=(b-a)^{-1}\int_a^bv(\xi)\ d\xi$? ...
2
votes
2answers
231 views

When is $\Vert \Vert f \Vert_p \Vert_q \leq \Vert\Vert f\Vert_q\Vert_p$?

Under which condition on two positive real numbers $p,q$ (not necessarily Hölder-conjugated) holds the following inequality for functions where the integral exists? $$\left(\int_{\Omega_1} \left( \...
1
vote
2answers
122 views

If $x+y+z=2$ prove that $\sum_{cyc}\frac{1}{\sqrt{x^2+y^2}}\ge2+\frac{1}{\sqrt{2}}$

Let $x,y,z$ be non-negative reals whose sum is $2$. Prove that $\frac{1}{\sqrt{x^2+y^2}}+\frac{1}{\sqrt{y^2+z^2}}+\frac{1}{\sqrt{z^2+x^2}}\ge2+\frac{1}{\sqrt{2}}$ I have tried bounding them up (...
1
vote
1answer
38 views

Integral of Function in $L^p(E)$ Times Functions in Dense Set Being $0$ Implies Function is $0$

Let $E$ be a measurable set, $1 \le p \le \infty$, $q$ the conjugate of $p$, and $S$ a dense subset of $L^q(E)$. If $g \in L^p(E)$ and $\int_E fg = 0$ for all $f \in S$, then $g=0$. I was able to ...
2
votes
1answer
47 views

By Holder inequality, show that $2\sin^2\lambda - \lambda \sin(2\lambda)\leq 2\lambda^2.$

Question: By using Holder's inequality, prove that for any real number $\lambda,$ $$4\sin^2\lambda - \lambda \sin(2\lambda)\leq 2\lambda^2.$$ Assume that $\lambda\geq 0.$ Since Holder's inequality ...
7
votes
1answer
85 views

product of $f(x)f(y)f(z)\geq1$ for $f(x)=ax^2+bx+c$ with $xyz=1$ and $a+b+c=1$.

Let $a,b,c,x,y,z$ be positive reals with $xyz=1$ and $a+b+c=1$. Let $f(x)=ax^2+bx+c$. Prove $f(x)f(y)f(z)\geq 1$. Im stuck, I tried a bunch of things but the best ive managed to obtain is: $f(x)f(y)...
1
vote
1answer
71 views

Prove this inequality $Σ_{cyc}\sqrt{\frac{a}{b+3c}}\ge \frac{3}{2}$ with $a;b;c>0$

Let $a,b,c>0$. Prove $$\sqrt{\frac{a}{b+3c}}+\sqrt{\frac{b}{c+3a}}+\sqrt{\frac{c}{a+3b}}\ge \frac{3}{2}$$ $A=\sqrt{\frac{a}{b+3c}}+\sqrt{\frac{b}{c+3a}}+\sqrt{\frac{c}{a+3b}}$ Holder: $A^2\cdot ...
2
votes
0answers
88 views

Is there a reverse Hölder inequality for matrices?

Let $A$ and $B$ be $n\times n$ matrices and let $p,q\in(1,\infty)$ such that $\frac{1}{p}+\frac{1}{q}=1$. Hölder's inequality for Schatten norms states that $$ \lvert \operatorname{Tr}(AB)\rvert \leq \...
3
votes
1answer
137 views

Inequality $a,b,c > 0 \Rightarrow \frac{b+c}{\sqrt{a^2 + bc}} + \frac{a+c}{\sqrt{b^2 + ac}} + \frac{a+b}{\sqrt{c^2 + ab}} \geq 4$

Prove: $a,b,c > 0 \Rightarrow \frac{b+c}{\sqrt{a^2 + bc}} + \frac{a+c}{\sqrt{b^2 + ac}} + \frac{a+b}{\sqrt{c^2 + ab}} \geq 4$ I was able to obtain a loose bound. My method: \begin{align} \frac{b+c}...
1
vote
0answers
60 views

Having trouble showing this inequality

Given the initial boundary value problem \begin{align*} &u_t = Du_{xx} + f(u), \quad 0<x<1, t>0 \\ &u(0,t) = u(1,t) = 0, \quad t>0 \\ &u(x,0) = u_{0}(x), \quad 0<x<1 \...
1
vote
1answer
44 views

Holders inequality for Hilbert Schmidt operators which are also trace class

Does Hilbert-Schmidt operators which are also trace class, satisfy Holders inequality? That is, we have two Hilbert Schmidt operators $A$ and $B$. Is the following true? $$\langle A, B \rangle \leq \...
1
vote
2answers
90 views

Prove this Hard inequality to CS or AM-GM?

Let $x,y,z\ge 0$ such $x+y+z=1$, show that $$\sqrt{\dfrac{x(x+1)}{1-x}}+\sqrt{\dfrac{y(y+1)}{1-y}}+\sqrt{\dfrac{z(z+1)}{1-z}}\ge\sqrt{6}$$ This inequality is creat by wangyongxi,and when $x=y=z=\...
0
votes
0answers
24 views

Dual of $p$-order cone

I'm attempting to prove that for $p\in[1,\infty]$ the dual of the cone $$K_p := \{x\in\mathbb{R}^n\,:\,\|\tilde{x}\|_p\le x_n\}$$ is $K_q$ with $\frac1q + \frac1p = 1$, where $\tilde{x} = (x_1,\ldots,...
1
vote
1answer
17 views

Weak convergence in $L^1$ with a bounded factor

Let $Z_n$ be a variable that converges weakly in $L^1$ to $Z$, that is for all $E\in\mathcal{F}$ it holds that $$ \mathbb{E}\left[Z_n\,I(E)\right]\to \mathbb{E}\left[Z\,I(E)\right]. $$ Let now $X$ ...
-1
votes
1answer
62 views

Application of Holder's Inequality: $\mathbb{E}|X| \ge \frac 1{\mathbb{E}\left[X^4\right]^{1/2}}$

Show that if $\mathbb{E}\!\left[X^2\right] =1$ and $\mathbb{E}\!\left[X^4\right] < \infty$, then $$\mathbb{E}|X| \ge \dfrac 1{\mathbb{E}\left[X^4\right]^{1/2}}$$ I have tried to write$X^2=|X|^r ...
0
votes
0answers
32 views

Average integration is increasing with measure

I was studying Evans' PDE, Morrey's Inequality. A step confused me: Is it correct that there is a constant depending only on $n$ so that $ \frac{\int_U f}{|U|}\leq C\frac{\int_V f}{|V|}$ if $U\subset ...
0
votes
1answer
83 views

L2 Norm Inequality

Let $f_1,f_2,f_3:\mathbb{R}^2\to\mathbb{R}_{\ge 0}$ be measurable, bounded, and compactly supported. Prove that $$\int_{\mathbb{R}^3}f_1(y,z)f_2(x,z)f_3(x,y)d(x,y,z)\le\lVert f_1\rVert_{L^2(\mathbb{R}...
1
vote
1answer
43 views

Question about Hölder's inequality proof

Let $1 < p,q < \infty$ with $\frac{1}{p} + \frac{1}{q}=1$. Then, for $a,b \in \mathbb{K}^n$, we have: $$|\sum_i a_ib_i| \leq \Vert a \Vert_p \Vert b \Vert_q$$ My book gives the proof in ...