I need to prove and inequality in Lp-spaces. Let (X,$\mathcal{S},\mu)$ and $1<p<+\infty$. Let $f:X\times X \rightarrow \mathbb{R}$ such that the y-section $f^{y}$ is $p$-integrable and further,
$$\int_{X} ||f^{y}||_{p} d\mu(y) < +\infty$$
Define, for all $x \in X$,
$$ g(x) = \int_{X} f(x,y)d\mu(y) $$.
I need to show that
$$ ||g||_p \le \int_{X} ||f^{y}||_{p} d\mu(y) $$
In other words,
$$\Bigg(\int_{X} \Bigg| \int_{X}f(x,y) d\mu(y) \Bigg|^{p} d\mu (x) \Bigg)^{1/p} \le \int_{X} \Bigg( \int_{X} |f(x,y)|^{p}d\mu(x) \Bigg)^{1/p} d\mu(y) $$
I was thinking in using Fubini's Theorem, and the fact that $\Big|\int_{X} f d\mu \Big| \le \int_{X} |f|d\mu$, but in general cases it doesn't seem to be useful.
Can somebody help me? :c
ADDED:
Acording with @Kabo Murphy 's answer, this is the Minkowski's Integral Inequality. The best proof I could find is here, but need some modifications: A kind of Minkowski inequality for integral
Following the last link, I was wondering:
a) Why does $||h||_{s'} = 1$ if $h \in L_{s'}[0,1]$?
b) Does it hold in general that $\int_{X}g(x)|h(x)|d\mu(x) = ||g||_{p}$ where $g(x) = \int_{X} f(x,y)d\mu(y)$ and $h(x) = g(x)^{p-1}||g||_{p}^{1-p}$?
Seems routine, but I couldn't find a proof :C
This is just Minkowski's inequality for integrals whose proof are easily available on the net. See https://en.wikipedia.org/wiki/Minkowski_inequality for example.