Some Ratio Monotonic Properties of a New Kind of Numbers introduced by Z.-W. Sun

TL;DR

This paper proves a conjecture by Z.-W. Sun on the ratio monotonicity of sequences related to a new class of numbers, using novel criteria for log-convexity and ratio log-concavity.

Contribution

It introduces new methods and criteria for analyzing ratio monotonicity and log-convexity, confirming Sun's conjecture on the properties of the numbers $S_n$.

Findings

Confirmed Sun's conjecture on ratio monotonicity.

Established new criteria for log-convexity and ratio log-concavity.

Applied interlacing method to analyze sequence properties.

Abstract

Recently, Z. W. Sun introduced a new kind of numbers $S_n$ and also posed a conjecture on ratio monotonicity of combinatorial sequences related to $S_n$. In this paper, by investigating some arithmetic properties of $S_n$, we give an affirmative answer to his conjecture. Our methods are based on a newly established criterion and interlacing method for log-convexity, and also the criterion for ratio log-concavity of a sequence due to Chen, Guo and Wang.