# 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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### Application of Hölder's inequality: $(a+b)^t \le 2^t(a^t+b^t)$ for $t\ge 1.$

While searching for a proof of the algebra property of Sobolev spaces ($H^s(\mathbb R^n)$ is an algebra when $s > n/2$), I found these notes. On page two the author states that if $t \ge 1$ then ...
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### Young’s inequality implies $a^ {n/(n+1)}\leq 2a+(1/n^2)$ for any $a>0$ and $n\geq 1$

I’ve seen this statement in a Yano’s article and I can not prove it. I take the Young’s inequality $a^{1/p}b^{1/q} \leq a/p + b/q$ where $1/p +1/q =1$. I’ve prove it in the case $a\geq 1$. In the ...
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### Hausdorff-Young inequality on T space

I know that this inequality work well on space $L^p(\mathbb{R})$. But is it possible to generalize this inequality to the $\mathbb{T}$ space? I think that on this space I can write this inequality ...
### Prove that $a b \leq \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$ [duplicate]
for any $a,b\in R^+$, how to prove that $$a b \leq \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$$ my try: if I can prove that $\frac{a^2+b^2}{2}\leq \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$then this ...