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 \(t\in N_a\).


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


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