Positivity

, Volume 22, Issue 2, pp 551–573 | Cite as

Monotonicity and non-monotonicity results for sequential fractional delta differences of mixed order

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Abstract

We consider the discrete fractional sequential difference \(\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\), where \(t\in \mathbb {N}_{3-\mu -\nu +a}\), in two separate cases, where in each case we require that \(\mu +\nu \in (1,2)\). In the first case, we show that when \(\mu \in (0,1)\) and \(\nu \in (1,2)\) it follows that the condition \(\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\ge 0\) implies that f is an increasing map when we impose that \(f(a)\ge 0\), \(\Delta f(a)\ge 0\), and \(\Delta f(a+1)\ge 0\). On the other hand, when \(\mu \in (1,2)\) and \(\nu \in (0,1)\) we demonstrate that the situation is very different and that this type of monotonicity result only holds when restricted to a proper subregion of the \((\mu ,\nu )\)-parameter space coupled with some additional auxiliary conditions.

Keywords

Discrete fractional calculus Monotonicity Sequential fractional delta difference 

Mathematics Subject Classification

Primary 26A48 39A70 39B62 Secondary 26A33 39A12 39A99 
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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsCreighton Preparatory SchoolOmahaUSA

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