Convexity for nabla and delta fractional differences.(English)Zbl 1320.39003

If $$(a,b)\in \mathbb R^2$$ and $$b>a$$, denote $N_a= \{a,a+1,a+2,\dots\},\;N^b_a= \{a,a+1, a+2,\dots, b\}.$ The authors prove results for nabla and delta fractional differences:
(1) If $$f: N_{a+1}\to \mathbb R$$ satisfies $$\nabla^\nu_a f(t)\geq 0$$ for each $$t\in N_{a+1}$$, with $$2\nu<3$$, then $$\nabla^2 f(t)\geq 0$$ for $$t\in N_{a+3}$$.
(2) If $$f: N_a\to \mathbb R$$ satisfies $$\nabla^\nu_a f(t)\geq 0$$ for each $$t\in N_a$$, with $$2<\nu<3$$, and $$f(a)\leq 0$$, $$\Delta f(a)\geq 0$$, $$\Delta^2 f(a)\geq 0$$, then $$\Delta f(t)\geq 0$$ for $$t\in N_a$$.
The authors consider the general case $$N-1<\nu<N$$ with $$N>3$$ and prove:
(3) $$\Delta^\nu_a f(t)\geq 0$$ for each $$t\in N_{a+1}$$, with $$N- 1<\nu< N$$, $$N\in N_1$$, $$(-1)^{N-i}\Delta^i f(a)\geq 0$$, $$0\leq i\leq N-2$$ and $$\Delta^{N-1}f(a)\geq 0$$, then $$\Delta^{N-1} f(t)\geq 0$$ for $$t\in N_a$$.

MSC:

 39A12 Discrete version of topics in analysis 39A70 Difference operators 26B12 Calculus of vector functions
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References:

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