Monotonicity and convexity for nabla fractional \(q\)-differences (Q2825626)

The paper starts from the context of fractional calculus; the nabla fractional sum is defined as NEWLINE\[NEWLINE\nabla^{-\nu}_{q,\rho(1)}f(t) := \int_{\rho(1)}^t(t-q^{-1}\tau) ^{(\nu-1)}_{(q-1)}f(\tau)\nabla_q\tau,\;t\in q^{N_0},\;f:q^{N_0}\to \mathbb R,\;q>0,\;\nu>0NEWLINE\]NEWLINE and the nabla \(q\)-fractional derivative of \(f\) at \(t\) as NEWLINE\[NEWLINE(\nabla^\nu_{q,\rho(1)})f(t) := (\nabla^m_q\nabla^{-(m-\nu)}_{q,\rho(1)}f) (t).NEWLINE\]NEWLINE The main results are incorporated in the followingNEWLINENEWLINETheorem A. Assume \(f:q^{N_0}\to \mathbb R\), \(\nabla^\nu_q f(t)\geq 0\) for each \(t\in q^{N_0}\), \(1<\nu<2\). Then \(\nabla_q f(t)\geq 0\), \(t\in q^{N_1}\).NEWLINENEWLINETheorem B. Assume \(f:q^{N_0}\to \mathbb R\), \(\nabla^\nu_q f(t)\geq 0\) for each \(t\in q^{N_1}\), \(2<\nu<3\). Then \(\nabla_q^2 f(t)\geq 0\), \(t\in q^{N_2}\).