$$A B C D B C + B C + C D + D A \lt 2 ( A C + B D ) ?$$
$\left\{\begin{matrix}D<-\frac{C\left(B-2A\right)}{A\left(BC\right)^{2}+A-2B+C}\text{, }&A>-\frac{C-2B}{B^{2}C^{2}+1}\\D\in \mathrm{R}\text{, }&\left(C>0\text{ and }A>\frac{B}{2}\right)\text{ or }\left(C<0\text{ and }A<\frac{B}{2}\right)\\D>-\frac{C\left(B-2A\right)}{A\left(BC\right)^{2}+A-2B+C}\text{, }&A<-\frac{C-2B}{B^{2}C^{2}+1}\end{matrix}\right.$
