The relationship between sequential fractional differences and convexity

Qualitative properties of nonlocal discrete operators

“…30, Theorem 7.17 See Theorem 5.6 and Section 6. We notice that our next theorem also improves those in the reference 38 for the sectors  3 ,  2 , and  4 .…”

“…The connection between the fractional difference operator () and convexity in the context of sequential fractional differences was first considered by Goodrich 38 . He consider, in relation to Figure 1, the cases of the open sectors and , named in Goodrich 38 : Cases I, II and III, respectively. Sharp convexity results for the sector can be found in Goodrich 42 …”

Convexity, monotonicity, and positivity results for sequential fractional nabla difference operators with discrete exponential kernels

“…42 In the case of nonsequential operators, these properties have been studied by Abdeljawad and Baleanu, 43 Atici and Uyanik, 44 Baoguo et al, 45 Dahal and Goodrich, 46 Du et al, 47 Goodrich, 48,49 Jia et al, 40,41,50 and Liu et al; 51 we remark also that the survey paper by Erbe et al 52 is a good reference for the nonsequential case, as is Chapter 7 of the textbook by Goodrich and Peterson. 53 On the other hand, in the case of sequential operators, these properties have been studied by Dahal and Goodrich, [54][55][56] Goodrich, [57][58][59][60][61] Goodrich and Lizama, 62,63 Goodrich et al, [64][65][66] and Goodrich and Muellner. 67 It turns out that the sequential case tends to be more qualitatively rich and interesting than the nonsequential case-see, for example, Goodrich, 60, Remark 2.…”

On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag–Leffler kernel

“…⇒ y is positive and/or monotone and/or convex, (1.3) where, once again, A is some nonnegative number. Some recent papers in this direction include those by Dahal and Goodrich, [39][40][41] Goodrich, [42][43][44][45][46] and Goodrich and Lizama. 37,47 An interesting and mathematically rich feature of such sequential operators is that there is a well documented and complex interaction between the magnitude of the orders 𝛼 and 𝛽 of the individual differences appearing in (1.3) and what, if anything, can be said about the qualitative behavior of the function y.…”

“…On the other hand, there are other papers that have studied a composition of fractional differences, which are usually known as “sequential fractional difference operators”—i.e., theorems that take the following form: where, once again, A is some nonnegative number. Some recent papers in this direction include those by Dahal and Goodrich, 39–41 Goodrich, 42–46 and Goodrich and Lizama 37,47 . An interesting and mathematically rich feature of such sequential operators is that there is a well documented and complex interaction between the magnitude of the orders α and β of the individual differences appearing in () and what, if anything, can be said about the qualitative behavior of the function y.…”