$$\begin{array}{l} \text { 19. Since }\left(\frac{a b}{c}+\frac{b c}{a}+\frac{c a}{b}\right)^{2}=\frac{a^{2} b^{2}}{c^{2}}+\frac{b^{2} c^{2}}{a^{2}}+\frac{c^{2} a^{2}}{b^{2}}+2\left(a^{2}+b^{2}+c^{2}\right)= \\ \frac{1}{2}\left(\frac{a^{2} b^{2}}{c^{2}}+\frac{c^{2} a^{2}}{b^{2}}\right)+\frac{1}{2}\left(\frac{b^{2} c^{2}}{a^{2}}+\frac{a^{2} b^{2}}{c^{2}}\right)+\frac{1}{2}\left(\frac{c^{2} a^{2}}{b^{2}}+\frac{b^{2} c^{2}}{a^{2}}\right)+2 \geqslant a^{2}+b^{2}+c^{2}+2= \end{array}$$
3. Therefore, the proposition holds.