We investigate relationships between the sign of the discrete fractional sequential difference $({\mathrm{\Delta }}_{1+a-\mu }^{\nu }{\mathrm{\Delta }}_a^{\mu }f)(t)$ and the convexity of the function $t\mapsto f(t)$. In particular, we consider the case in which the bound $({\mathrm{\Delta }}_{1+a-\mu }^{\nu }{\mathrm{\Delta }}_a^{\mu }f)(t)\ge \epsilon f(a),$for some $\epsilon >0$ and where $f(a)<0$, is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation.

我们研究离散分数顺序差的正负号之间的关系 $（{\mathrm{\Delta }}_{\mathrm{1个}+\mathrm{一种}-\mu }^{\nu }{\mathrm{\Delta }}_{\mathrm{一种}}^{\mu }F）（Ť）$ 和函数的凸性 $Ť\mapsto F（Ť）$。特别是，我们考虑了$（{\mathrm{\Delta }}_{\mathrm{1个}+\mathrm{一种}-\mu }^{\nu }{\mathrm{\Delta }}_{\mathrm{一种}}^{\mu }F）（Ť）\ge \epsilon F（\mathrm{一种}），$对于一些 $\epsilon >0$ 在哪里 $F（\mathrm{一种}）<0$，很满意。因此，我们考虑了顺序差可能为负的情况，并且我们表明，即使分数差可以为负，但上述不等式仍可以隐含函数f的凸性。这证明了小数案例和非小数案例之间的显着差异。我们结合使用了硬分析和数值模拟。