Some positive results for exponential-kernel difference operators of Riemann-Liouville type

We established positivity of $\mathrm{\nabla }f$ obtained from a systematic computation of a composition of sequential fractional differences of the function $f$ that satisfy certain conditions in a negative lower bound setup. First, we considered the different order sequential fractional differences in which we need a complicated condition. Next, we equalled the order of fractional differences and we saw that a simpler condition will be needed. We illustrated our positivity results for an increasing function of the rising type.