Monotonicity results for delta fractional differences revisited

Year, pages: 2017, 895 - 906

In this paper, by means of a recently obtained inequality, we study the delta fractional difference, and we obtain the following interrelated theorems, which improve recent results in the literature. \noindent{\bf \textsc{Theorem A}.} \textit{Assume that $f:\N_a\to\R$ and that $Δ^ν_af(t)≥ 0$, for each $t\in\N_a+2-ν$, with $1 <ν < 2$. If $f(a+1)≥ ((ν) / (k+2))f(a)$, for each $k\in\N₀$, then $Δ f(t)≥ 0$ for $t\in\N_a+1$}. \noindent{\bf \textsc{Theorem B}.} \textit{Assume that $f:\N_a\to\R$ and that $Δ^ν_af(t)≥0$, for each $t\in\N_a+2-ν$, with $1 < ν < 2$. If

$$ f(a+2)≥((ν) / (k+1))f(a+1)+(((k+1-ν)ν) / ((k+2)(k+3)))f(a) $$

for each $k\in\N₁$, then $Δ f(t)≥ 0$ for $t\in\N_a+2$.} \noindent {\bf \textsc{Theorem C}.} \textit{Assume that $f:\N_a\to\R$ and that $Δ^ν_af(t)≥0$, for each $t\in\N_a+2-ν$, with $1< ν < 2$. If

$$ f(a+3)≥ ((ν) / (k))f(a+2)+(((k-ν)ν) / (k(k+1)))f(a+1)+(((k+1-ν)(k-ν)ν) / ((k+2)(k+1)k))f(a) $$

for $k\in\N₂$, then $Δ f(t)≥ 0$, for $t\in\N_a+3$.} \noindent In addition, we obtain the following result, which extends a recent result due to Atici and Uyanik. \noindent{\bf \textsc{Theorem D}.} \textit{Assume that $f:\N_a\rightarrow\R$, $Δ^Nf(t)≥ 0$ for $t\in\N_a$, and $(-1)^N-iΔⁱf(a)≤ 0$ for $i=0,1,…,N-1$. Then $Δ^ν_af(t)≥ 0$ for $t\in\N_a+N-ν$.}

Erbe, L., Goodrich, C., Jia, B., Peterson, A. 2017. Monotonicity results for delta fractional differences revisited. In Mathematica Slovaca, vol. 67, no.4, pp. 895-906. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0018

Erbe, L., Goodrich, C., Jia, B., Peterson, A. (2017). Monotonicity results for delta fractional differences revisited. Mathematica Slovaca, 67(4), 895-906. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0018