Lp spaces and Hölder Inequality

Question

I have a question about Lp Spaces and Hölder Inequality.

My question is;

$f, g \in L_{3}(\mathbb{R})$ and $\|f\|_{L_{3}(\mathbb{R})}=\|g\|_{L_{3}(\mathbb{R})}=2$ .

$$ \left|\int_{\mathbb{R}} f^{2}(x) g(x) d x\right| \leq ? $$

Here my opinion is using the Hölder's Inequality in this question. Here's a link about it. https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality i know that there are also different versions of this inequality.i looked at some books. but i dont know how to apply this inequality on this question. Thanks for your answers.

Answer 1

$\int |f|^2|g|\leq\Big((\int|f|^3\Big)^{2/3}\big(\int|g|^3\Big)^{1/3}=\|f\|^2_3\|g\|_3=2^22=8$