Holder Inequality when $0 < p < 1$

Question

If $0 < p < 1$, $f \in L^p$, and $\int \lvert g \rvert^q < \infty$, show that $$\int \lvert fg \rvert \ge (\int \lvert f \rvert^p)^{\frac{1}{p}}(\int \lvert g \rvert^q)^{\frac{1}{q}}$$ My feeling is that I can use $\frac{p-q}{p}$ and $\frac{q-p}{q}$ as a conjugate pair, and apply Holder Inequality to reach such a conclusion. But I tried a lot and failed to get the desired inequality. I was considering to prove $(\int \lvert fg \rvert )(\int \lvert g \rvert^q)^{-\frac{1}{q}} \ge (\int \lvert f \rvert^p)^{\frac{1}{p}}$, then $(\int \lvert fg \rvert )^{p-q}(\int \lvert g \rvert^q)^{-\frac{p-q}{q}} \ge (\int \lvert f \rvert^p)^{\frac{p-q}{p}}$, i.e. $(\int \lvert fg \rvert )^{p-q}(\int \lvert g \rvert^q)^{\frac{q-p}{q}} \ge (\int \lvert f \rvert^p)^{\frac{p-q}{p}}$. Seems like dead ends to me..