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Monotonicity results for delta fractional differences revisited

  • Lynn Erbe , Christopher S. Goodrich , Baoguo Jia und Allan Peterson EMAIL logo
Veröffentlicht/Copyright: 14. Juli 2017
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 67 Heft 4

Abstract

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.

Theorem A

Assume that f : ℕa → ℝ and thatΔaνf(t) ≥ 0, for eacht ∈ ℕa+2−ν, with 1 < ν < 2. Iff(a+1)≥νk+2f(a),for each k ∈ ℕ0, then Δ f(t) ≥ 0 for t ∈ ℕa+1.

Theorem B

Assume that f : ℕa → ℝ and thatΔaνf(t) ≥ 0, for each t ∈ ℕa+2−ν, with 1 < ν < 2. If

f(a+2)≥νk+1f(a+1)+(k+1−ν)ν(k+2)(k+3)f(a)
for each k ∈ ℕ1, then Δ f(t) ≥ 0 for t ∈ ℕa+2.

Theorem C

Assume that f : ℕa → ℝ and thatΔaνf(t) ≥ 0, for each t ∈ ℕa+2−ν, with 1 < ν < 2. If

f(a+3)≥νkf(a+2)+(k−ν)νk(k+1)f(a+1)+(k+1−ν)(k−ν)ν(k+2)(k+1)kf(a)
for k ∈ ℕ2, then Δ f(t) ≥ 0, fort ∈ ℕa+3.

In addition, we obtain the following result, which extends a recent result due to Atici and Uyanik.

Theorem D

Assume that f : ℕa → ℝ, ΔNf(t) ≥ 0 for t ∈ ℕa, and (−1)N−iΔif(a) ≤ 0 fori = 0, 1, …, N − 1. ThenΔaνf(t) ≥ 0 fort ∈ ℕa+N−ν.

MSC 2010: Primary 26A33; 26A48; 39A12; Secondary 39A70; 39A99
Keywords: Delta fractional difference; Taylor monomial; monotonicity

(Communicated by Ján Borsík)

The third author, B. Jia, was supported by The National Natural Science Foundation of China (No.11271380) and Guangdong Province Key Laboratory of Computational Science.

Acknowledgement

The authors would like to thank the two anonymous referees for their useful suggestions and comments.

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Received: 2015-6-1
Accepted: 2016-1-22
Published Online: 2017-7-14
Published in Print: 2017-8-28

© 2017 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0018
Schlagwörter für diesen Artikel
Delta fractional difference; Taylor monomial; monotonicity

Artikel in diesem Heft

  2. Outer measure on effect algebras
  4. Hamiltonian ordered algebras and congruence extension
  6. Automorphism groups with some finiteness conditions
  8. Only finitely many Tribonacci Diophantine triples exist
  10. Abundant semigroups with a *-normal idempotent
  12. Stability results for fractional differential equations with state-dependent delay and not instantaneous impulses
  14. Monotonicity results for delta fractional differences revisited
  16. M-Cantorvals of Ferens type
  18. Comparison of ψ-porous topologies
  20. On certain generalized matrix methods of convergence in (ℓ)-groups
  22. Some applications of first-order differential subordinations
  24. Hankel determinant for a class of analytic functions involving conical domains defined by subordination
  26. Nonoscillation and exponential stability of the second order delay differential equation with damping
  28. Positive solutions of perturbed nonlinear hammerstein integral equation
  30. Ricci solitons on 3-dimensional cosymplectic manifolds
  32. Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups
  34. The super socle of the ring of continuous functions
  36. On the internal approach to differential equations 2. The controllability structure
  38. Finiteness of the discrete spectrum in a three-body system with point interaction
  40. On a subclass of Bazilevic functions
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