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We prove that a class of convexity-type results for sequential fractional delta differences is sharp. More precisely, we consider the sequential fractional delta difference map t↦Δ1-μ+aνΔaμf(t), for t∈N3+a-μ-ν, where 0

Consider the following ν-th order Caputo delta fractional equation (0.1) The following asymptotic results are obtained. Theorem A. Assume0 0. Then the solutions of the equation (0.1) withx(a) > 0satisfy Theorem B. Assume0 0satisfy This shows that the solutions of the Caputo delta fractional equation , with x(a) > 0 have similar asymptotic behavior with the solutions of the first-order delta difference equation Δx(t) = cx(t),c >− 1. Copyright © 2016 John Wiley & Sons, Ltd.

We demonstrate that some recent results regarding the connection between the convexity of the map t → f (t) and the sign of Δa f (t) , with 2 < ν < 3 , can be improved. In particular, by utilizing a recent inequality due to Jia, Erbe, and Peterson, we are able to improve some of the existing results in the literature. As part of this study we illustrate the improvements that our results afford by providing several specific examples of their application. Mathematics subject classification (2010): Primary: 26A51, 39A12, 39B62; Secondary: 26A33, 39A70, 39A99.