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- Open Access
On a ratio monotonicity conjecture of a new kind of numbers
- Brian Yi Sun^{1}Email author
https://doi.org/10.1186/s13660-018-1614-1
© The Author(s) 2018
- Received: 27 August 2017
- Accepted: 12 January 2018
- Published: 25 January 2018
Abstract
It is known that the concept of ratio monotonicity is closely related to log-convexity and log-concavity. In this paper, by exploring the log-behavior properties of a new combinatorial sequence defined by Z.-W. Sun, we completely solve a conjecture on ratio monotonicity by him.
Keywords
- log-concavity
- log-convexity
- ratio monotonicity
- interlacing method
MSC
- 05A20
- 05A10
- 11B65
- 11B37
1 Introduction
To be self-contained in this paper, let us first review some necessary and important concepts.
Clearly, a sequence \(\{z_{n}\}_{n=0}^{\infty}\) is (strictly) log-convex (resp. log-concave) if and only if the sequence \(\{z_{n+1}/z_{n}\} _{n\geq0}\) is (strictly) increasing (resp. decreasing). So, to study the ratio monotonicity is equivalent to study the log-convexity and log-concavity; see [1].
In recent years, Sun [2, 3] posed a series of conjectures on monotonicity of sequences of the forms \(\{z_{n+1}/z_{n}\}_{n\geq0}^{\infty}\), \(\{\sqrt[n]{z_{n}}\}_{n\geq1}\), and \(\{\sqrt[n+1]{z_{n+1}}/\sqrt [n]{z_{n}}\}_{n\geq1}\). It is worth mentioning that many scholars have made valuable progress on this topic, such as Chen et al. [4], Hou et al. [5], Luca and Stănică [6], Wang an Zhu [1], Sun et al. [7], and Zhao [8].
He conjectured that the sequence \(\{R_{n}\}_{n=0}^{\infty}\) is strictly ratio increasing to the limit \(3+2\sqrt{2}\) and that the sequence \(\{ \sqrt[n]{R_{n}}\}_{n=1}^{\infty}\) is strictly ratio decreasing to the limit 1.
In this paper, by studying the log-behavior properties of the sequence \(\{R_{n}\}_{n=0}^{\infty}\) we completely solve the ratio monotonicity conjecture on \(\{R_{n}\}_{n=0}^{\infty}\).
Theorem 1.1
The sequence \(\{R_{n+1}/R_{n}\} _{n=3}^{\infty}\) is strictly increasing to the limit \(3+2\sqrt{2}\), and the sequence \(\{\sqrt[n+1]{R_{n+1}}/\sqrt[n]{R_{n}}\}_{n=5}^{\infty}\) is strictly decreasing to the limit 1.
In what follows, in Section 2 we first introduce the interlacing method which can be used to verify log-behavior property of a sequence. In Section 3 we establish a lower bound and an upper bound for \(R_{n+1}/R_{n}\). We will give and prove some limits and log-behavior properties related to the sequence \(\{R_{n}\}_{n=0}^{\infty}\) in Section 4 and finally prove Theorem 1.1 therein. In the end, we conclude this paper with some open conjectures for further research.
2 The interlacing method
The interlacing method can be found in [13], yet it was formally considered as a method to solve logarithmic behavior of combinatorial sequences by Dos̆lić and Veljan [14], in which it was also called the sandwich method.
Proposition 2.1
3 Bounds for \(R_{n+1}/R_{n}\)
In this section, we establish lower and upper bounds for \(R_{n+1}/R_{n}\).
Lemma 3.1
Proof
The proof of \(r_{k+2}>b_{k+2}\) is similar, so we omit it for brevity.
Remark 3.2
This bound was found by a lot of computer experiments. It is interesting to explore a unified method that can be used to find lower and upper bounds for the sequence \(\{\frac{z_{n+1}}{z_{n}}\}_{n\geq0}\), where \(\{z_{n}\}_{n\geq0}\) is a sequence satisfying a four-term recurrence.
4 Log-behavior of the sequence \(\{R_{n}\}_{n=0}^{\infty}\)
In this section, some log-behavior and limits properties can be deduced by using Lemma 3.1.
Theorem 4.1
The sequence \(\{R_{n}\}_{n=4}^{\infty}\) is strictly log-convex. Equivalently, the sequence \(\{R_{n+1}/R_{n}\}_{n=3}^{\infty}\) is strictly increasing.
Proof
Corollary 4.2
Theorem 4.3
Proof
This completes the proof. □
Theorem 4.4
Proof
Theorem 4.5
The sequence \(\{\sqrt[n]{R_{n}}\}_{n=5}^{\infty}\) is strictly log-concave. Equivalently, the sequence \(\{\frac{\sqrt[n+1]{R_{n+1}}}{\sqrt [n]{R_{n}}}\}_{n=5}^{\infty}\) is strictly decreasing.
Before giving the proof of Theorem 4.5, we have to use to a criterion for log-concavity of sequences in the form of \(\{\sqrt[n]{z_{n}}\}_{n=1}^{\infty}\); this criterion was established by Xia [16].
Theorem 4.6
- (i)
\(0< f(n)<\frac{z_{n}}{z_{n-1}}<f(n+1)\);
- (ii)
\(\frac{f(n+1)}{f(n+3)}>1-\frac{k_{0}}{n^{2}+n+2}\);
- (iii)
\((1-\frac{k_{0}}{N_{0}^{2}+N_{0}+2} )^{N_{0}^{2}+N_{0}+2} f^{2N_{0}}(N_{0})>z_{N_{0}}^{2}\).
We are now in a position to prove Theorem 4.5.
Proof of Theorem 4.5
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1
By Theorems 4.1 and 4.2 we confirm the first part of Theorem 1.1. Moreover, Theorems 4.5 and 4.4 imply the second part of Theorem 1.1. This ends the proof. □
We conclude the paper with some conjectures for further research.
Conjecture 4.7
The sequence \(\{\frac{R_{n+1}}{R_{n}}\}_{n\geq4}\) is log-concave, that is, \(R_{n}\) is ratio log-concave for \(n\geq4\).
Conjecture 4.8
The sequence \(\{R_{n}^{2}-R_{n+1}R_{n-1}\}_{n\geq6}\) is ∞-log-concave.
Declarations
Acknowledgements
We wish to give many thanks to the referee for helpful suggestions and comments, which greatly helped to improve the presentation of this paper. This work was partially supported by the China Postdoctoral Science Foundation (No. 2017M621188) and the National Science Foundation of China (Nos. 11701491 and 11726630).
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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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