On Ratio Monotonicity of a New Kind of Numbers Conjectured by Z.-W. Sun

Abstract

Recently, Z. W. Sun put forward a series of conjectures on monotonicity of combinatorial sequences in the form of $\{z_n/z_{n-1}{\}}_{n=N}^{\infty }$ and $\{\sqrt[n+1]{z_{n+1}}/\sqrt[n]{z_n}{\}}_{n=N}^{\infty }$ for some positive integer $N$, where $\{z_n{\}}_{n=0}^{\infty }$ is a sequence of positive integers. Luca and Stănică, Hou et al., Chen et al., Sun and Yang proved some of them. In this paper, we give an affirmative answer to monotonicity of another new kind of number conjectured by Z. W. Sun via interlacing method for log-convexity and log-concavity of a sequence, and we also use the criterion for log-concavity of a sequence in the form of $\{\sqrt[n]{z_n}{\}}_{n=1}^{\infty }$ due to Xia.

draft,13 pages. arXiv admin note: text overlap with arXiv:1512.01010

References in corpus (6)

• Log-concavity and LC-positivity
• Log-balanced combinatorial sequences
• Two new kinds of numbers and related divisibility results
• Higher order log-monotonicity of combinatorial sequences
• Sun's log-concavity conjecture on the Catalan-Larcombe-French sequence
• Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums