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Proof of a conjecture of ZW Sun on ratio monotonicity
Journal of Inequalities and Applications volume 2016, Article number: 272 (2016)
Abstract
In this paper, we study the logbehavior of a new sequence \(\{S_{n}\} _{n=0}^{\infty}\), which was defined by ZW Sun. We find that the sequence is logconvex by using the interlacing method. Additionally, we consider ratio logbehavior 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 logconcave. Our results give an affirmative answer to a conjecture of ZW Sun on the ratio monotonicity of this new sequence.
Introduction
Throughout the paper, we denote by \(\mathbb{N}\) the set of nonnegative integers. The main objective of this paper aims to confirm a conjecture on ratio monotonicity of a new kind of sequence \(\{S_{n}\}_{n=0}^{\infty}\) via studying its logbehavior properties. The sequence \(\{S_{n}\}_{n=0}^{\infty}\) was introduced by Sun in [1, 2] and defined as follows:
Sun studied congruence and divisibility properties of this kind of numbers in [1, 2] and posed the following conjecture.
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.
To begin with, let us review some related concepts. Let \(\{z_{n}\} _{n=0}^{\infty}\) be a sequence of positive real numbers. We say a sequence \(\{z_{n}\}_{n= 0}^{\infty}\) is (strictly) ratio monotonic if its ratio sequence \(\{\frac{z_{n+1}}{z_{n}}\}_{n=0}^{\infty}\) is (strictly) monotonically increasing or (strictly) decreasing as n increases. A sequence \(\{z_{n}\}_{n= 0}^{\infty}\) is said to be logconvex (resp. logconcave) if, for all \(n\geq1\),
Meanwhile, the sequence \(\{z_{n}\}_{n=0}^{\infty}\) is called strictly logconvex (resp. logconcave) if the inequality in (1.2) is strict for \(n\geq n_{0}\) for some \(n_{0}\in\mathbb{N}\). Indeed, ratio monotonicity is equivalent to logbehavior. According to the definitions, it is easy to see that a ratio monotonically increasing (resp. decreasing) sequence is itself a logconvex (resp. logconcave) sequence and vice versa.
So far, many criteria for logbehavior 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 logbehavior in order to study the logbehavior of sequences of the form \(\{\sqrt[n]{z_{n}}\}_{n=1}^{\infty}\). By ratio logconcavity (resp. logconvexity) of a sequence \(\{z_{n}\}_{n=0}^{\infty}\), we mean that the sequence \(\{\frac{z_{n+1}}{z_{n}}\}_{n=0}^{\infty}\) is logconcave (resp. logconvex). They found that the ratio logconcavity (resp. logconvexity) 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 logconcave (resp. logconvex) if it satisfies certain initial conditions; see [15], Theorem 3.1 and Theorem 3.6. To make this paper selfcontained, we will recall their criteria in Section 3.
The main results of the present paper can be stated as follows.
Theorem 1.2
The sequence \(\{S_{n}\}_{n=0}^{\infty}\) is strictly logconvex, that is,
Theorem 1.3
The sequence \(\{S_{n}\}_{n=0}^{\infty}\) is ratio logconcave, that is,
Theorem 1.4
The sequence \(\{\sqrt[n]{S_{n}}\}_{n=1}^{\infty}\) is strictly logconcave.
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 threeterm recurrence for \(S_{n}\), a lower and upper bound for \(\frac {S_{n}}{S_{n1}}\). In Section 3, we give proofs of our main theorems.
Preliminaries
A threeterm recurrence
The Zeilberger algorithm [16] yields the following fourterm recurrence for \(S_{n}\):
This recurrence cannot easily be tackled as almost all criteria for logbehavior are concerned with threeterm recurrences. So it is indispensable for us to find a threeterm recurrence. The following lemma was first obtained awkwardly by solving a linear system of equations. Afterwards, we found it can be deduced from a threeterm recurrence for \(4n S_{n}\), which was established by a technique way due to Guo and Liu [17].
Lemma 2.1
Let \(S_{n}\) be defined in (1.1). Then it satisfies a threeterm recurrence:
Proof
Let \(u_{n}=4nS_{n}\). Guo and Liu [17], Eq. (2.4), found a threeterm recurrence for \(u_{n}\), i.e.,
Substituting \(4nS_{n}\) for \(u_{n}\) in (2.2) and then simplifying, the recurrence (2.1) follows easily. □
Bounds for \(\frac{S_{n}}{S_{n1}}\)
In [4], Chen and Xia provided a heuristic approach to find bounds for \(\frac{z_{n}}{z_{n1}}\), where \(z_{n}\) satisfies a threeterm recurrence. The following bounds can be acquired by using their method.
Lemma 2.2
Let
Then we have
Proof
We proceed our proof by induction on n. For the sake of simplicity, let
To begin with,
so inequality (2.3) holds for \(n=2\).
Suppose that \(h(n1)< s_{n}< h(n)\), we proceed to show that \(h(n)< s_{n+1}< h(n+1)\).
On the one hand, by Lemma 2.1, we have
which obviously implies \(s_{n+1}< h(n+1)\).
On the other hand, consider that, for \(n\geq1\),
Evidently, this gives us \(s_{n+1}>h(n)\).
According to an inductive argument, it follows that, for all \(n\geq2\), we have
□
As a corollary, we have the following.
Corollary 2.3
Let \(S_{n}\) be defined by (1.1). Then we have
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]
Suppose that \(\{z_{n}\}_{n=0}^{\infty}\) is a sequence of positive numbers. Then, for some positive integer N, the sequence \(\{z_{n}\}_{n= N}^{\infty}\) is logconvex (resp. logconcave) if there exists an increasing (resp. a decreasing) sequence \(\{h(n)\}_{n=0}^{\infty}\) such that
holds for \(n\geq N+1\), where \(q_{n}=\frac{z_{n+1}}{z_{n}}\). Moreover, the sequence \(\{z_{n}\}_{n= N}^{\infty}\) is strictly logconvex (resp. strictly logconcave) if and only if the above inequalities (3.1) are strict.
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
Let \(\{z_{n}\}_{n=0}^{\infty}\) be the sequence defined by the following recurrence:
Assume that \(v(n)<0\) for \(n\geq2\). If there exist a nonnegative integer N and a function \(h(n)\) such that, for all \(n\geq N+2\),

(i)
\(\frac{3u(n)}{4}\leq\frac{z_{n}}{z_{n1}}\leq h(n)\);

(ii)
\(h(n)^{4}u(n)h(n)^{3}u(n+1)v(n)h(n)v(n)v(n+1)<0\),
then \(\{z_{n}\}_{n=N}^{\infty}\) is ratio logconcave.
Theorem 3.3
[15], Theorem 3.6
Assume that k is a positive integer. If a sequence \(\{z_{n}\} _{n=k}^{\infty}\) is ratio logconcave and
then the sequence \(\{\sqrt[n]{z_{n}}\}_{n=k}^{\infty}\) is strictly logconcave.
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 logconvex 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
The sequence \(\{\sqrt[n]{S_{n}}\}_{n=1}^{\infty}\) is strictly increasing. Moreover,
Proof
By Corollary 3.4, it follows that
Consider that \(S_{0}=1\), so
which implies
This is equivalent to
that is,
Additionally, consider that, for a real sequence \(\{z_{n}\}_{n=1}^{\infty}\) of positive real numbers, it was shown that
and
see Rudin [18]. The inequalities in (3.3) and (3.4) imply that
if \(\lim_{n\rightarrow\infty}\frac{z_{n}}{z_{n1}}\) exists. By Corollary 2.3, we arrive at (3.2).
This completes the proof. □
Proof of Theorem 1.3
By Lemma 2.1, the recurrence (2.1) implies that
To keep the notation in Theorem 3.2, here we still let
Consider that
which shows that
Additionally, for \(n\geq1\),
where
Combining the inequalities (3.5) and (3.6), we arrive at our statement in Theorem 1.3 by Theorem 3.2. □
Remark 3.6
Notice that the first author of the present paper and Zhao [19] also found some criteria for ratio logbehavior, which can also be used to prove ratio logconcavity of \(\{S_{n}\}_{n=0}^{\infty}\).
Proof of Theorem 1.4
By Theorem 3.3, it suffices to find a positive integer k such that
Let \(k=1\), we have
since
Therefore, by Theorem 1.3 and Theorem 3.3, it follows that \(\{\sqrt[n]{S_{n}}\}_{n=1}^{\infty}\) is strictly logconcave. □
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
For \(S_{n}\), we have
Proof
With the aid of Lemma 2.2, we have
Therefore, we can deduce that
and
Resorting to Mathematica 10.0, we find that
Thus we can arrive at
which implies
This finishes the proof. □
Proof of Theorem 1.5
The first part of Conjecture 1.1 follows from Theorem 1.2, Corollary 2.3 and Corollary 3.4. The second part follows from Theorem 1.4, Corollary 3.7, and Corollary 3.8. This completes the proof of Theorem 1.5. □
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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).
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Sun, B.Y., Hu, Y. & Wu, B. Proof of a conjecture of ZW Sun on ratio monotonicity. J Inequal Appl 2016, 272 (2016). https://doi.org/10.1186/s136600161221y
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DOI: https://doi.org/10.1186/s136600161221y
MSC
 05A20
 05A10
 11B65
 11B37
Keywords
 logconvexity
 logconcavity
 ratio monotonicity
 interlacing method
 ratio logconcavity