Doubt in variant of Holder inequality

Question

Let $p, q, r \in [1, \infty), r \neq \infty$ such that $1/p+1/q=1/r$. If $f \in {L(X)} ^ p$ and $g \in {L (X)}^q$. Is it true that $|f|^{p/r}<|f|^p$? According to I do not, because if I consider $f = 1/2$ constant and $p = 4 = q$ and $r = 2$ all in space $X = [0,1]$ then $ \int |f|^4 <\infty$ but $|1/2|^2 <|1/2|^ {4}$ something absurd. It is right?