# Questions tagged [holder-inequality]

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

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### Applying Holder Inequality to obtain a estimate.

I'm reading a paper concerned with PDE, a inequality in it makes me confuse. We have known: $$\int|\nabla u|^{p-2}|A|^2\eta^2\leq C\int|\nabla u|^{p}$$ and author says we can obtain following ...
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### I want to prove for all $a > 0$, $b > 0$, $\frac{2ab}{a + b} + \sqrt{\frac{1}{2}\left(a^2 + b^2\right)} \geq \frac{a + b}{2} + \sqrt{ab}$ [closed]

How can I prove that for all $a > 0$ and $b > 0$, $$\frac{2ab}{a + b} + \sqrt{\frac{1}{2}\left(a^2 + b^2\right)} \geq \frac{a + b}{2} + \sqrt{ab}$$ I found this inequality in an mathematic ...
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### Pulling out a maximum term from an integral

Suppose $g$ is nonnegative on $[0,1]$. Is the following true? $$\left| \int_0^1 f(x) g(x) dx\right| \leq \|g\|_\infty \left| \int_0^1 f(x) dx\right|$$ This would be like Holder's inequality, except ...
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### Bounding growth of generalised mean $M_p(a)$ as $p$ increases

This is a follow-up question to the one previously asked here. on the geometric growth rate of generalized mean. For some $p$ and positive numbers $a=(a_1,\dots,a_n)\in\mathbb{R}^{+}$, let $M_p(a)$ ...
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### relationship between generalized mean ratios $M_4(a)/M_2(a)$ vs $M_2(a)/M_1(a)$

For some $p$ and positive numbers $a=(a_1,\dots,a_n)\in\mathbb{R}^{+}$, let $M_p(a)$ denote the generalised or Hölder mean, defined as \begin{align} M_p = \left(\frac1n\sum_{i=1}^n a_i^p \right)^{1/p},...
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### How to bound $m(A)^{-1}\int_A|f(x) - c|^tdx$ above by $\left(m(A)^{-1}\int_A|f(x) - c|dx\right)^t$ for $0 < t < 1$ and $A$ a compact set

Let $m$ denote the Lebesgue measure on $\mathbb{R}^n$, $f$ be a locally integrable function in $\mathbb{R}^n$, let $0 < t < 1$, $A\subset\mathbb{R}^n$ be a compact subset and $c$ some finite ...
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### Intuition for proofs of Minkowski Inequality

I've always found the usual proofs of both Minkowski's and Hölder's inequalities extremely unintuitive and unsatisfying. It is very unclear how someone who wanted to prove Minkowski's inequality would ...
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### Help with Holder's Inequality [closed]

I was studying Holder's inequality and I came across the second problem used at the 2001 IMO because it involved Holder. The question I want to ask is what is the reasoning behind the first line? ...
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### For 1≤p<q≤$\infty$, show that $L^{p}(\mathbb{Z},\mu) \subset L^{q}(\mathbb{Z},\mu)$, $\mu$ being the counting measure. [duplicate]

I have to show the inclusion $L^{p}(\mathbb{Z},\mu) \subset L^{q}(\mathbb{Z},\mu)$ with the counting measure. Using 1≤p<q≤$\infty$ and the definition of the $L^{p}/L^{q}$ spaces I have tried to ...
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### For $x,y,z∈ℝ^{+}$,without using Hölder's inequality prove that $\frac{x}{\sqrt{x^2+8yz}}+\frac{y}{\sqrt{y^2+8xz}}+\frac{z}{\sqrt{z^2+8xy}}\geq1$.

For $x,y,z∈ℝ^{+}$, prove that $\frac{x}{\sqrt{x^2+8yz}}+\frac{y}{\sqrt{y^2+8xz}}+\frac{z}{\sqrt{z^2+8xy}}\geq1$. In this question solution used Hölder's inequality, but I am looking a solution ... 50 views

### Hölder Condition and equicontinuity

My question is the following: If $|f_n(x)-f_n(y)|≤M|x-y|^\alpha$ for some fixed M and $\alpha>0$ and all x,y in a compact interval, show that ${f_n}$ is uniformly equicontinuous. As a criterion for ...
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### Proof of Holder's Inequality in Multivariable Calculus

I am self studying Calculus 3 (Multivariate/Vector Calculus) as a hobby and I came across this question. I graduated with a degree in chemistry, so I know a bit of math, but not too much to answer ...
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### Product of power means inequality: prove $M_0^{\frac{n}{n+k}}M_k^{\frac{k}{n+k}} \leq M_1$

Inspired by this question Prove that $\frac{{x_1}^2+{x_2}^2+\cdots+{x_n}^2}{n}x_1x_2\cdots x_n\le\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)^{n+2}$, I wonder if the following is true: Let $k > 0$, ...
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