I need some help with the following problem
Let $1 < p < q <\infty$, $f \in L^p(\mu)\cap L^p(\mu)$ and $r \in [p,q]$. Show that there exists a $c \in [0,1]$ such that $$||f||_r \leq ||f||_p^{1-c} \cdot ||f||_q^c$$
We can decompose $r=(1-c)p+cq$ for some $c \in [0,1]$. Then we can use the Hölder inequality for $$||f||_r^r=\int |f|^r = \int |f|^{(1-c)p}|f|^{cq} \leq \left(\int |f|^p\right)^{1-c}\left(\int |f|^q\right)^{c}$$ So we get $$||f||^r_r \leq (||f||_p^p)^{1-c}(||f||_q^q)^c$$ Now I would take the $r$-th root. However the exponents of the seminorms do not really match for any other upper bound.