Analytical and Numerical Monotonicity Results for Discrete Fractional Sequential Differences with Negative Lower Bound

We investigate the relationship between the sign of the discrete fractional sequential difference \begin{document}$ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $\end{document} and the monotonicity of the function \begin{document}$ t\mapsto f(t) $\end{document} . More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound \begin{document}$ \begin{equation} \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t)\ge-\varepsilon f(a),\notag \end{equation} $\end{document} for some \begin{document}$ \varepsilon>0 $\end{document} . We prove that even though the fractional difference can be negative, the monotonicity of the function \begin{document}$ f $\end{document} , nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.