Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

TL;DR

This paper introduces new techniques to analyze the monotonicity of specific number-theoretic and combinatorial sequences, successfully verifying several conjectures posed by Sun and others.

Contribution

It develops unified methods to prove monotonicity properties and confirms multiple conjectures in the field.

Findings

Verified several conjectures on sequence monotonicity

Developed techniques applicable to number-theoretic sequences

Provided a unified approach for different conjectures

Abstract

We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.