19. In an acute triangle $\triangle A B C$, $a<b<c$, let $p=\frac{1}{2}(a+b+c), q=a \cos A+b \cos B+c \cos C$, try to compare the sizes of $p, q$.


19. Since $a\cos B>\cos C$, therefore $q$ is the reverse order product, by Chebyshev's inequality
$$
\frac{1}{3} q \leqslant \frac{1}{3}(a+b+c) \cdot \frac{1}{3}(\cos A+\cos B+\cos C) \leqslant \frac{1}{3} p,
$$

i.e., $q \leqslant p$.