# Questions tagged [young-inequality]

This tag is for questions relating to Young's inequality, a special case of the weighted AM-GM inequality. It is very useful in real analysis, including as a tool to prove Hölder's inequality. It is also a special case of a more general inequality known as Young's inequality for increasing functions.

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### Young's Inequality in Locally Convex Spaces

I'm trying to understand a proof of a Lemma that's in relation to Young's inequality, the lemma is the following : Suppose $\frac{1}{p} + \frac{1}{q} = 1$ Let $a,b>0$ and let $1<p<\infty$ ...
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### Are there any refinements of $\langle \vec x,\vec y\rangle \leq \lVert \vec x\rVert_1\lVert \vec y\rVert_\infty$?

The standard Young's inequality states, for $a,b>0$ and $\nu\in[0,1]$, that $$\nu a + (1-\nu)b\geq a^{\nu}b^{1-\nu}.$$ This has been refined in various ways, for example see the refined Young's ...
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### Does Young's inequality hold only for conjugate exponents?

Suppose that $ab \leq \frac{1}{p}a^p+\frac{1}{q}b^q$ holds for every real numbers $a,b\ge 0$. (where $p,q>0$ are some fixed numbers). Is it true that $\frac{1}{p}+\frac{1}{q}=1$? I guess so, and ...
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### 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 \...
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### Connecting Young's Inequality for Increasing Functions w/Young's Inequality for Conjugate Holder Exponents

From the Wikipedia page on Young's Inequality: This above statement of Young's Inequality is the most frequent one that I've encountered in textbooks. Yet consider: This second version ...
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Something doesn't completely makes sense to me in Minkowski's inequality and I'm trying to understand it. All throughout let $a_i, b_i \geq 0$, and $p > 1$, $\frac{1}{p} + \frac{1}{q} = 1$. ...