For an arrangement of $n$ lines in the real projective plane, we denote by $f$ the number of regions into which the real projective plane is divided by the lines. Using Bojanowski's inequality, we establish a new lower bound for $f$. In particular, we show that if no more than $\frac{2}{3}n$ lines intersect at any point, then $f \ge \frac{1}{6}n^{2}$