Certain embedding theorems

We obtain necessary and sufficient conditions such that, for $f(x)$ from $L^p(0,1)$, the integral
$$ \int_0^1|f(x)|^q\,dx\quad(0<p<1,\quad p<q<p(1-p)^{-1}) $$
is convergent, or for $f\in L^p[0,1]$ for all $p\ge1$, the integral $\int_0^1e^{|f(x)|}\,dx$ is convergent.