Monotonicity rule for the quotient of two functions and its application

Zhen Hang Yang, Wei Mao Qian, Yu Ming Chu, Wen Zhang

Research output: Contribution to journalArticlepeer-review

58 Scopus citations

Abstract

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.

Original languageEnglish
Article number106
JournalJournal of Inequalities and Applications
Volume2017
Issue number1
DOIs
StatePublished - 2017
Externally publishedYes

Keywords

  • absolute error
  • complete elliptic integral
  • monotonicity rule
  • relative error

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