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In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels (a−1ABR∇δ,γy)(η) of order 0<δ<0.5, β=1, 0<γ≤1 starting at a−1. If (a−1ABR∇δ,γy)(η)≥0, then we deduce that y(η) is δ2γ-increasing. That is, y(η+1)≥δ2γy(η) for each η∈Na:=a,a+1,…. Conversely, if y(η) is increasing with y(a)≥0, then we deduce that (a−1ABR∇δ,γy)(η)≥0. Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.

Consider the multiple testing problem of testing k null hypotheses, where the unknown family of distributions is assumed to satisfy a certain monotonicity assumption. Attention is restricted to procedures that control the familywise error rate in the strong sense and which satisfy a monotonicity condition. Under these assumptions, we prove certain maximin optimality results for some well-known stepdown and stepup procedures.

In this paper we will discuss constraints on the number of (non-dummy) players & on the distribution of votes such that local monotonicity is satisfied for the Public Good Index. These results are compared to properties that are related to constraints on the redistribution of votes (such as implied by global monotonicity). The discussion shows that monotonicity is not a straightforward criterion of classification for power measures. 2 Figures, 1 Appendix, 20 References. Adapted from the source document.