Monotonicity rules for the ratio of power series

TL;DR

This paper develops new monotonicity rules for the ratio of two power series, especially when the sequence of their coefficient ratios changes monotonicity twice, and introduces a local monotonicity rule.

Contribution

It provides novel monotonicity criteria for power series ratios under complex coefficient ratio behaviors and introduces a local monotonicity rule.

Findings

New monotonicity rules for power series ratios.

Conditions for changing monotonicity of coefficient ratios.

Introduction of a local monotonicity rule.

Abstract

In this paper, we present some monotonicity rules for the ratio of two power series $x\mapsto \sum_{k=0}^\infty a_k x^k / \sum_{k=0}^\infty b_k x^k$ under the assumption that the monotonicity of the sequence ${a_k/b_k}$ changes twice. Additionally, we introduce a local monotonicity rule in this paper.