I thought the integral version of this equality is $ \int_a^b |f(x)g(x)| dx \le (b-a)\cdot \int_{a}^b |f(x)|dx \cdot \int_{a}^b |g(x)|dx$ for real continuous function $f,g$ on $[a,b]$.
But I found that there is a counterexample of this inequality.
Is there a similar inequality with respect to integral corresponding to $\sum_i |a_i||b_i| \le (\sum_i |a_i|) (\sum_i |b_i|)$?