Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then
$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$
I refer to $(*)$ as the Liapunov's Inequality for $L_p$ spaces. I've been told that it is possible to prove this inequality by "an appropiate application" of the regular Hölder inequality (that is, $||fg||_1 \leq ||f||_p ||g||_q$, where $p$ and $q$ are conjugate exponents). I know how to prove $(*)$ using other inequalities closely related to the regular Hölder inequality, but not that inequality itself. After awhile playing with the regular Hölder inequality, I haven't been able to get $(*)$. So, can someone illuminate me with the "appropiate application" of the regular Hölder inequality?