Proof of a conjecture of Z-W Sun on ratio monotonicity

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 log-behavior 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.

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 log-convex (resp. log-concave) if, for all \(n\geq1\), Meanwhile, the sequence \(\{z_{n}\}_{n=0}^{\infty}\) is called strictly log-convex (resp. log-concave) 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 log-behavior. According to the definitions, it is easy to see that a ratio monotonically increasing (resp. decreasing) sequence is itself a log-convex (resp. log-concave) sequence and vice versa.

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.

On the basis of Theorem 1.2 and Theorem 1.4, we can conclude the following result.

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.

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].

To prove Theorem 1.3 and Theorem 1.4, the following criteria due to Chen et al. [15] are also indispensable.

We are now in a position to prove our main theorems.

As a corollary, we have the following.

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.