Questions tagged [holder-inequality]

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

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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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2 answers
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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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1 answer
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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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3 votes
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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: $$...
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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 ...
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4 votes
2 answers
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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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6 votes
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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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3 votes
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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 ...
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2 votes
1 answer
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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 ...
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5 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 ...
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2 votes
1 answer
31 views

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 ...
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1 vote
2 answers
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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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2 answers
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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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  • 373
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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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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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  • 1,035
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0 answers
28 views

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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2 votes
1 answer
32 views

Two implications of an operator that preserves positivity on L2

Suppose $(X,\mathcal{A},m)$ is a probability space. Let A be an operator on function spaces. I am currently taking a course in measure theory, and in the lecture slides the following two remarks are ...
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1 vote
1 answer
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Scalar Product with Telescope Sum

I have a basic analysis problem that I don't manage to solve: Let $(a_n)_{n \in \mathbb{N}}\subseteq \mathbb{R}$ be a strictly positive sequence with $a_n \overset{n \to \infty}{\longrightarrow}\infty$...
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1 vote
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Is the following estimate of this integral correct?

Let $\epsilon>0$ and suppose that $U_t^\epsilon, V_t^\epsilon, \Gamma^{2\epsilon}_t$ are some real functions of time (depending on $\epsilon$) defined on $[0,T]$, $T < \infty$ and assume that ...
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0 votes
0 answers
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Array power-means / generalisation of Hölder inequality

Now cross-posted to MO. If you have a comment or answer, please post it there instead. Summary: I have a generalisation of Hölder's inequality that I can neither prove nor find references to. ...
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1 vote
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Is $d(x, y)={(\sum_{i=1}^{n}{|{x_i} -{y_i}|}^p})^{1/p}$ , $x, y\in\mathbb{R^n}$, $0<p<1$ metric on $\mathbb{R^n}$?

$X=\mathbb{R^n}$ Define , $d:X×X\rightarrow\mathbb{R}$ by $d(x, y)={(\sum_{i=1}^{n}{|{x_i} -{y_i}|}^p})^{1/p}$ , $x, y\in\mathbb{R^n}$, $0<p<1$ Question: Is $d$ a metric on $\mathbb{R^n}$? ...
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2 votes
1 answer
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Is true $\int\limits_{-\infty}^{\infty} f(t)\,g(t)\,dt \leq \sqrt{\int\limits_{-\infty}^{\infty} |f(t)|^2 dt}\,\cdot \sup\limits_t|g(t)|$?

Does holds $\int\limits_{-\infty}^{\infty} f(t)\,g(t)\,dt \leq \sqrt{\int\limits_{-\infty}^{\infty} |f(t)|^2 dt}\,\cdot \sup\limits_t|g(t)|$ true for every real valued functions $f(t),\, g(t)$? I want ...
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1 vote
0 answers
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Prove $\sum_{k=1}^n a_kb_k \leq \left (\sum_{k=1}^n a_k^p \right)^{1/p} \left (\sum_{k=1}^n b_k^{q} \right)^{1/q}$ without Hölder's inequality

I have proved this version by applying Hölder's inequality . Could you confirm if my proof is correct? I also would like to ask for a non-measure-theoretic proof. For $p,q \ge 0$ such that $1/p+1/q=1$...
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2 votes
2 answers
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A generalized version of Hölder's inequality

I've found this version from this Wikipedia page. I've re-written the proof to make my understanding clear. Could you confirm if my attempt is correct? Let $(X, \mathcal A, \mu)$ be a $\sigma$-...
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0 votes
1 answer
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Hölder condition for the function $f(x)=x^{\frac{1}{n}}$ [closed]

I think my algebra is not very good.. I'm trying to show that $f:[0,\infty)\to[0,\infty)$ defined as $f(x)=x^{\frac{1}{n}}$ ,$n\in\mathbb{N}$ satisfies the Hölder condition $|f(x)-f(y)|\leq c|x-y|^{\...
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1 vote
0 answers
91 views

Baby rudin theorem 8.18

$\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt.$ Here is the Hölder's inequality: (1/p) + (1/q) = 1. |$\int_a^bfg$ $d\alpha$| $\leq$ {$\int_a^b$ $|f|^p$ $d\alpha$}$^{1/p}$ {$\int_a^b$ $|g|^q$ $d\alpha$}$^{...
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1 vote
1 answer
60 views

Show $E(|X|) \ge \frac{1}{\sqrt{E(X^4)}}$

We are asked to show the following: $E(|X|) \ge \frac{1}{\sqrt{E(X^4)}}$ Given that $E(X^2) = 1$ and $E(X^4) < \infty$ Using Holder's inequality I can show that $E(X^4) \le 1$ assuming $X,Y = X^2,...
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1 vote
0 answers
68 views

It is true that $L^p(\mathbb{R}^n)\subset \mathscr{S}'(\mathbb{R}^n)$, $1\leq p\leq\infty$?

Remember that $\mathscr{S}'$ is the space of tempered distributions. In a certain text they suggest that this statement is true, and that he uses Holder's inequality to prove it. The question is, how ...
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1 vote
2 answers
97 views

Show the $d_p(x,y)$ is not a metric on $\mathbb{R}^2$

Question: For $0\lt p\lt 1$ and $x, y \in\mathbb{R}^2$ with $x=(\xi_1, \xi_2), y = (\eta_1, \eta_2)$, let $d :\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R} $ Be defined as $d_p(x, y) = (|\...
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2 votes
3 answers
156 views

Prove that $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \ge ab + bc + ca$

For all positive $a,b,c $ satisfying $a+b+c = 3$,Prove: $$ \sum_{cyc} \sqrt[3]{a} \ge \sum_{cyc} ab $$ This is a hard problem and I tried it myself, but it's really hard without using advanced ...
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5 votes
1 answer
107 views

Inequality with strange sum of cubic roots

For positive numbers $a$, $b$, $c \geq 0$ and $a+b+c=1$ show that: $\sqrt[3]{4+17a^2b}+\sqrt[3]{4+17b^2c}+\sqrt[3]{4+17c^2a}+10 \Big(\frac{1}{27}-abc \Big) \geq 5$ I tried to use $AM-GM$ with $\frac{...
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2 votes
1 answer
100 views

Understanding the intuition behind a proof of Holder’s Inequality

I’m reading a book on Linear Algebra and one of the exercises is to prove Holder’s inequality. In the exercise, the author actually breaks the question in several subitens I order to help the reader. ...
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