# Questions tagged [holder-inequality]

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

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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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### Holder inequality Proof Explanation

I don't understand why $$ptf(x)h(x)^{p-1}\leq (h(x)+tf(x))^p-h(x)^p$$.
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### 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 ...
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### 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 $\alpha \in [0,1]$ ...
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### 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 ...