Monotonicity rules for the ratio of power series with applications

In this paper, we present monotonicity rules for the ratio of two power series x ↦ {∑ₖ₌₀∞ a_k x^k}{∑ₖ₌₀∞ b_k x^k} under the assumption that the sequence \{a_k/b_k\}k ≥ 0 changes monotonicity twice. We also prove that the number of monotonicity changes of this function does not exceed that of the sequence \{a_k/b_k\}k ≥ 0 . As an application, we establish a necessary and sufficient condition for the monotonicity of a function involving the confluent hypergeometric function of the first kind, x ↦ {e⁻ˣ}{x+1} M(a, b, cx) , where a, b, c > 0 .