Monotonicity rules for the ratio of power series

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