Monotonicity rule for the quotient of two functions and its application

Abstract In the article, we provide a monotonicity rule for the function [ P ( x ) + A ( x ) ] / [ P ( x ) + B ( x ) ] $[P(x)+A(x)]/[P(x)+B(x)]$ , where P ( x ) $P(x)$ is a positive differentiable and decreasing function defined on ( − R , R ) $(-R, R)$ ( R > 0 $R>0$ ), and A ( x ) = ∑ n = n 0 ∞ a n x n $A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$ and B ( x ) = ∑ n = n 0 ∞ b n x n $B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$ are two real power series converging on ( − R , R ) $(-R, R)$ such that the sequence { a n / b n } n = n 0 ∞ $\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$ is increasing (decreasing) with a n 0 / b n 0 ≥ ( ≤ ) 1 $a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$ and b n > 0 $b_{n}>0$ for all n ≥ n 0 $n\geq n_{0}$ . As applications, we present new bounds for the complete elliptic integral E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 t d t $\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ ( 0 < r < 1 $0< r