Proof of a conjecture of Z-W Sun on ratio monotonicity
- Brian Yi Sun^{1}Email author,
- Yingying Hu^{1} and
- Baoyindureng Wu^{1}
https://doi.org/10.1186/s13660-016-1221-y
© Sun et al. 2016
Received: 5 August 2016
Accepted: 25 October 2016
Published: 4 November 2016
Abstract
In this paper, we study the log-behavior of a new sequence \(\{S_{n}\} _{n=0}^{\infty}\), which was defined by Z-W Sun. We find that the sequence is log-convex by using the interlacing method. Additionally, we consider ratio log-behavior of \(\{S_{n}\}_{n=0}^{\infty}\) and find the sequences \(\{S_{n+1}/S_{n}\}_{n=0}^{\infty}\) and \(\{\sqrt[n]{S_{n}}\} _{n=1}^{\infty}\) are log-concave. Our results give an affirmative answer to a conjecture of Z-W Sun on the ratio monotonicity of this new sequence.
Keywords
log-convexity log-concavity ratio monotonicity interlacing method ratio log-concavityMSC
05A20 05A10 11B65 11B371 Introduction
Conjecture 1.1
[1], Conjecture 5.2(ii), [2], Conjecture 4.4
The sequence \(\{\frac{S_{n+1}}{S_{n}}\}_{n=3}^{\infty}\) is strictly increasing to the limit 9, and the sequence \(\{\frac{\sqrt [n+1]{S_{n+1}}}{\sqrt[n]{S_{n}}}\}_{n=1}^{\infty}\) is strictly decreasing to the limit 1.
So far, many criteria for log-behavior of a sequence have been developed; see [3–10] and the references therein for details. Also, there have been some important progress on ratio monotonicity since many conjectures on ratio monotonicity were posed by Sun [11]. For example, the reader may refer to [12–14]. Recently, Chen et al. [15] introduced a notion called ratio log-behavior in order to study the log-behavior of sequences of the form \(\{\sqrt[n]{z_{n}}\}_{n=1}^{\infty}\). By ratio log-concavity (resp. log-convexity) of a sequence \(\{z_{n}\}_{n=0}^{\infty}\), we mean that the sequence \(\{\frac{z_{n+1}}{z_{n}}\}_{n=0}^{\infty}\) is log-concave (resp. log-convex). They found that the ratio log-concavity (resp. log-convexity) of a positive sequence \(\{z_{n}\}_{n=k_{0}}^{\infty}\) for some positive integer \(k_{0}\) can imply the sequence \(\{\sqrt[n]{z_{n}}\} _{n=k_{0}}^{\infty}\) is strictly log-concave (resp. log-convex) if it satisfies certain initial conditions; see [15], Theorem 3.1 and Theorem 3.6. To make this paper self-contained, we will recall their criteria in Section 3.
The main results of the present paper can be stated as follows.
Theorem 1.2
Theorem 1.3
Theorem 1.4
The sequence \(\{\sqrt[n]{S_{n}}\}_{n=1}^{\infty}\) is strictly log-concave.
On the basis of Theorem 1.2 and Theorem 1.4, we can conclude the following result.
Theorem 1.5
Conjecture 1.1 is true.
The remainder of the paper is organized as follows. We give some preliminaries work in Section 2, including a three-term recurrence for \(S_{n}\), a lower and upper bound for \(\frac {S_{n}}{S_{n-1}}\). In Section 3, we give proofs of our main theorems.
2 Preliminaries
2.1 A three-term recurrence
Lemma 2.1
Proof
2.2 Bounds for \(\frac{S_{n}}{S_{n-1}}\)
In [4], Chen and Xia provided a heuristic approach to find bounds for \(\frac{z_{n}}{z_{n-1}}\), where \(z_{n}\) satisfies a three-term recurrence. The following bounds can be acquired by using their method.
Lemma 2.2
Proof
Suppose that \(h(n-1)< s_{n}< h(n)\), we proceed to show that \(h(n)< s_{n+1}< h(n+1)\).
As a corollary, we have the following.
Corollary 2.3
3 Proofs of theorems
Before giving proofs of our theorems, we need to recall some known results. The following proposition first appeared in [6] and is formally called the interlacing method by Došlić and Veljan [5].
Proposition 3.1
[5]
To prove Theorem 1.3 and Theorem 1.4, the following criteria due to Chen et al. [15] are also indispensable.
Theorem 3.2
[15], Theorem 3.1
- (i)
\(\frac{3u(n)}{4}\leq\frac{z_{n}}{z_{n-1}}\leq h(n)\);
- (ii)
\(h(n)^{4}-u(n)h(n)^{3}-u(n+1)v(n)h(n)-v(n)v(n+1)<0\),
Theorem 3.3
[15], Theorem 3.6
We are now in a position to prove our main theorems.
Proof of Theorem 1.2
Since \(h(n)\) is strictly monotonically increasing, it follows that \(\{ S_{n}\}_{n=1}^{\infty}\) is strictly log-convex by Lemma 2.2 and Proposition 3.1. □
As a corollary, we have the following.
Corollary 3.4
The sequence \(\{\frac{S_{n+1}}{S_{n}}\}_{n=1}^{\infty}\) is strictly monotonically increasing.
Since \(\{S_{n}\}_{n=0}^{\infty}\) is a positive sequence, we can define the sequence \(\{\sqrt[n]{S_{n}}\}_{n=1}^{\infty}\). Then we have the following result.
Corollary 3.5
Proof
This completes the proof. □
Proof of Theorem 1.3
Remark 3.6
Notice that the first author of the present paper and Zhao [19] also found some criteria for ratio log-behavior, which can also be used to prove ratio log-concavity of \(\{S_{n}\}_{n=0}^{\infty}\).
Proof of Theorem 1.4
Corollary 3.7
The sequence \(\{\frac{\sqrt[n+1]{S_{n+1}}}{\sqrt[n]{S_{n}}}\}_{n=1}^{\infty}\) is strictly monotonically decreasing.
Corollary 3.8
Proof
Declarations
Acknowledgements
We would like to thank the Editor, the Associate Editor and the anonymous referees for their careful reading and constructive comments which have helped us to significantly improve the presentation of the paper. The first two authors was supported by the Scientific Research Program of the Higher Education Institution of Xinjiang Uygur Autonomous Region (No. XJEDU2016S032) and the third author was supported by NSFC (No. 11571294).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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