A New Method for Refining the Shafer’s Equality and Bounding the Definite Integrals

  • Xiao-Diao Chen
  • Song Jin
  • Chen Li-Geng
  • Yigang WangEmail author


This paper presents an interpolation-based method for bounding some smooth functions, including the arctangent functions related to Shafer’s inequality. Given the form of new bounding functions, the interpolation technique is also utilized for determining the corresponding unknown coefficients, and the resulting functions bound the given function very well under some preset condition. Two applications are shown, one is to refine Shafer’s inequality, and the other is to approximate the definite integrals of some special functions; both of them have wide applications in computer science, mathematics, physical sciences and engineering. Experimental results show that the new bounds achieve much better bounds than those of prevailing methods.


Shafer’s inequality arctangent function approximations and estimations bounded bounds definite integral 

Mathematics Subject Classification

Primary 26A09 42A10 Secondary 26D99 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Complex Systems Modeling and SimulationHangzhou Dianzi UniversityHangzhouPeople’s Republic of China

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