New monotonicity conditions in discrete fractional calculus with applications to extremality conditions
Abstract We present some initial results for detecting extrema of a function f : ℕ a → ℝ {f:\mathbb{N}_{a}\rightarrow\mathbb{R}} when the only a priori information is some limited pointwise data about f as well as information about an appropriate fractional difference of f. In particular, such results provide some analogues of the well-known tests for detecting local extrema using the sign of Δ ⁢ f ⁢ ( t ) {\Delta f(t)} . Certain of these results are consequences of some new monotonicity conditions, which relate conditions on the sign of Δ a ν ⁢ f ⁢ ( t ) {\Delta_{a}^{\nu}f(t)} , for 0 < ν < 1 {0<\nu<1} , to the monotonicity of f. Finally, we provide some numerical examples to illustrate the results.