Two classes of power mappings with boomerang uniformity 2

Let \begin{document}$ q $\end{document} be an odd prime power. Let \begin{document}$ F_1(x) = x^{d_1} $\end{document} and \begin{document}$ F_2(x) = x^{d_2} $\end{document} be power mappings over \begin{document}$ \mathrm{GF}(q^2) $\end{document}, where \begin{document}$ d_1 = q-1 $\end{document} and \begin{document}$ d_2 = d_1+\frac{q^2-1}{2} = \frac{(q-1)(q+3)}{2} $\end{document}. In this paper, we study the boomerang uniformity of \begin{document}$ F_1 $\end{document} and \begin{document}$ F_2 $\end{document} via their differential properties. It is shown that the boomerang uniformity of \begin{document}$ F_i $\end{document} (\begin{document}$ i = 1,2 $\end{document}) is 2 with some conditions on \begin{document}$ q $\end{document}.