Inequality in $L^p(\Omega)$ space with $\Omega$ $\sigma$-finite.

Question

Let $f_{1},f_{2},...,f_{k}$ be k functions such that $f_{i} \in L^{p_{i}}(\Omega)\hspace5mm\forall i $ with $1\leq p_{i}\leq \infty$ and $\sum_{i=1}^{k}\frac{1}{p_{i}}\leq 1$. Set $f(x)=\prod_{i=1}^{k}f_{i}(x). $ Prove that $f\in L^p(\Omega)$ with $\frac{1}{p}=\sum_{i=1}^{k}\frac{1}{p_{i}}$ and that $||f||_{p}\leq \prod_{i=1}^{k}||f||_{p_{i}}.$ \ I got answer for p=1 by simply apply generalized holders inequality but not able to prove for $p<1$ I tried for this but didnt got any clue how to do.