Liapunov's Inequality for $L_p$ spaces
Write $$\int |f|^rd\mu=\int\color{green}{|f|^{p\lambda}}\color{red}{|f|^{q(1-\lambda)}}d\mu,$$ then apply Hölder's inequality to the exponent $\frac 1{\lambda}>1$ (its conjugate exponent is $\frac 1{1-\lambda}$). This gives $$\int |f|^rd\mu\leqslant \color{green}{\left(\int |f|^p\right)^{\lambda}}\color{red}{\left(\int |f|^q\right)^{1-\lambda}}=\color{green}{\lVert f\rVert_p^{p\lambda}}\color{red}{\lVert f\rVert_q^{q(1-\lambda)}},$$ what is wanted.