Monotonicity rule for the quotient of two functions and its application

• Published in: Journal of inequalities and applications Vol. 2017; no. 1; p. 106
• Main Authors: Yang, Zhen-Hang, Qian, Wei-Mao, Chu, Yu-Ming, Zhang, Wen
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2017; Springer Nature B.V; SpringerOpen
• Subjects: absolute error; Analysis; Applications of Mathematics; complete elliptic integral; Mathematics; Mathematics and Statistics; monotonicity rule; Power series; relative error; relative error; 26A48; 33E05; monotonicity rule; absolute error; complete elliptic integral
• ISSN: 1025-5834; 1029-242X; 1029-242X
• DOI: 10.1186/s13660-017-1383-2

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 = n 0 ∞ a n x n and B ( x ) = ∑ n = n 0 ∞ b n x n are two real power series converging on ( − R , R ) such that the sequence { a n / b n } n = n 0 ∞ is increasing (decreasing) with a n 0 / b n 0 ≥ ( ≤ ) 1 and b n > 0 for all n ≥ 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 ( 0 < r < 1 ) of the second kind.