Monotonicity rule for the quotient of two functions and its application

摘要

In the article, we provide a monotonicity rule for the function [P(x) + A(x)]/[P(x) + B(x)], where P(x) is a positive differentiable and decreasing function defined on (−R, R) (R  0), and A(x)=∑n=n0∞anxn and B(x)=∑n=n0∞bnxn are two real power series converging on (−R, R) such that the sequence {an/bn}n=n0∞ is increasing (decreasing) with an0/bn0 ≥ ( ≤ ) 1 and bn  0 for all n ≥ n0. As applications, we present new bounds for the complete elliptic integral E(r)=∫0π/21−r2sin2tdt (0  r  1) of the second kind.