Monotonicity rules for the ratio of power series

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.