Monotonicity results for delta fractional differences revisited

Lynn Erbe; Christopher S. Goodrich; Baoguo Jia; Allan Peterson

doi:10.1515/ms-2017-0018

Published by De Gruyter July 14, 2017

Monotonicity results for delta fractional differences revisited

Lynn Erbe , Christopher S. Goodrich , Baoguo Jia and Allan Peterson

From the journal Mathematica Slovaca

https://doi.org/10.1515/ms-2017-0018

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

Theorem B

Assume that f : ℕ_a → ℝ and thatΔaνf(t) ≥ 0, for each t ∈ ℕ_a+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 ∈ ℕ_a+2.

Theorem C

Assume that f : ℕ_a → ℝ and thatΔaνf(t) ≥ 0, for each t ∈ ℕ_a+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 ∈ ℕ_a+3.

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

Theorem D

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

Monotonicity results for delta fractional differences revisited

Abstract

Theorem A

Theorem B

Theorem C

Theorem D

Acknowledgement

References

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