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.

Highlights

  • To be self-contained in this paper, let us first review some necessary and important concepts.Let {zn}n≥0 be a number-theoretic or combinatorial sequence of positive numbers

  • Sun [2] posed a conjecture on ratio monotonicity of the sequence n n n+k 1

  • The ratio monotonicity conjecture on the sequence {Rn}∞ n=0 can not be attacked with the methods of Sun et al [7] since there exists no three-term recurrence for Rn

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Summary

Introduction

To be self-contained in this paper, let us first review some necessary and important concepts.Let {zn}n≥0 be a number-theoretic or combinatorial sequence of positive numbers. A sequence {zn}∞ n=0 is called log-convex If the inequality in (1.1) is strict, we call the sequence {zn}∞ n=0 strictly log-convex A sequence {zn}∞ n=0 is (strictly) log-convex

Results
Conclusion

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