On additive complements in natural numbers

We prove that for additive complements $A$ and $B$, if $$\underset{x\to +\mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{A(x)B(x)}{x}<\frac{4}{3},$$ then $$A(x)B(x)-x\to +\mathrm{\infty }(x\to +\mathrm{\infty })\mathrm{.}$$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.