Monotonicity rule for the quotient of two functions and its application

In the article, we provide a monotonicity rule for the function [Formula: see text], where [Formula: see text] is a positive differentiable and decreasing function defined on [Formula: see text] ([Formula: see text]), and [Formula: see text] and [Formula: see text] are two real power series converging on [Formula: see text] such that the sequence [Formula: see text] is increasing (decreasing) with [Formula: see text] and [Formula: see text] for all [Formula: see text]. As applications, we present new bounds for the complete elliptic integral [Formula: see text] ([Formula: see text]) of the second kind.