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Questions tagged [holder-inequality]

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

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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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3answers
75 views

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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1answer
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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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1answer
25 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
109 views

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
43 views

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
34 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
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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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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
28 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
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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
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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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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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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
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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 $p,q &...
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$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}} < ∞ \}....
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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^...
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0answers
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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 ...
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1answer
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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
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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 < \...
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91 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 ...
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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}}<\...
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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 ...
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1answer
123 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}+\...
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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 ...
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1answer
49 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) + (...
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1answer
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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 $\...
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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 $\...
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4answers
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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 ...
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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 ...
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1answer
21 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]...
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2answers
187 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}\...
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2answers
62 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}...
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1answer
34 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 ...
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2answers
58 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|...
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1answer
144 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 ...
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1answer
52 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 ...
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1answer
63 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 ...
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0answers
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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 ...
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1answer
39 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: $$((x_1+x_2)^4+(y_1+y_2)^4)^{1/4}\le (x_1^4+y_1^4)^{1/4}+(x_2^4+y_2^4)^{1/4}$$
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1answer
399 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 $\frac1p+\frac1q=1,~p, q>1$. ...
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1answer
74 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 $$ (a+b)^{n-1}(A+B) \geq ((a^{n-1}A)^{1/n} + (b^{n-1}B)^{1/n})^n $$ ...