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The relationship between sequential fractional differences and convexity

Goodrich Christopher S. (Creighton Preparatory School, Department of Mathematics, Omaha, USA)

We consider the relationship between the sign of the sequential fractional difference Δν2+a-μΔμa f(t), t  N4+a-μ-ν, and the convexity of the map f : Na → R in the case where μ  (1,2), ν  (1,2), and μ + ν  (2,3). In particular, we demonstrate that there exist dissimilarities between the sequential case studied here and the non-sequential case, which has been previously studied. In addition, we describe a fundamental inequality that Δ2f(t) must satisfy whenever Δν2+a-μΔaf(t)≥ 0. This inequality also reveals some dissimilarities between the sequential and non-sequential settings. We provide some numerical examples to illustrate the results. Finally, in addition to the case μ (1,2), ν (1,2), and μ+ν (2,3), we also consider results in both the case μ  (0,1),ν (1,2), and μ + ν  (2,3) as well as the case μ  (1,2), ν (0,1), and μ + ν  (2,3).

Keywords: discrete fractional calculus, convexity, concavity, sequential fractional delta difference, Taylor monomial