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I have to show that for $1le p < q <infty$

neitheir
$L^p((0, infty),mathbb{C}) subset L^q((0, infty),mathbb{C})$

nor
$L^q((0, infty),mathbb{C}) subset L^p((0, infty),mathbb{C})$.

I previously had to show, that there is no $a in mathbb{R}$ with which the function f:$(0, infty) to mathbb{R}$ f(x)=x^a

would be lebesgue integrable and there is a hint that i should use this information, but i have no idea how to do so.

I would be very thankfull for any kind of help.

Try this: if $p<q$ pick $p<r<q$ and consider the function $f:(0,+infty)tomathbb{R}$ given by
$$
f(x)=x^{-frac{1}{r}}chi_{(0,1)}(x)
$$
it's easy to show that $fin L^p(0,+infty)$ but $fnotin L^q(0,+infty)$. For the second part you might just take
$$
g(x)=x^{-frac{1}{r}}chi_{(1,+infty)}(x)
$$
Here $fin L^q(0,+infty)$ but $fnotin L^p(0,+infty)$. This example should show you that in general for $p<q$ you have $L^qsubset L^p$ if the space you are considering have "measure mass condensed in bounded sets" (not formal, but in a suitable sense it works), for example in $[0,1]$. If the mass is "condensed at $infty$", for example with $mathbb{N}$ and the counting measure, then is reversed: $ell^psubsetell^q$.