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Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 180–189 | Cite as

A double inequality for an integral mean in terms of the exponential and logarithmic means

  • Feng QiEmail author
  • Xiao-Ting Shi
  • Fang-Fang Liu
  • Zhen-Hang Yang
Article

Abstract

In the paper, the authors aim to present a double inequality for the integral mean
$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }a^{\cos ^2\theta }b^{\sin ^2\theta }{{\mathrm{d}}}\theta \end{aligned}$$
in terms of the exponential and logarithmic means. For attaining the goal, by the Cauchy residue theorem in the theory of complex functions and properties of definite integrals, the authors represent the above integral mean in terms of the modified Bessel function of the first kind. Finally, by virtue of inequalities for the hyperbolic tangent function, the authors further refine upper bounds in the newly-established double inequality in terms of the arithmetic and geometric means.

Keywords

Exponential mean Logarithmic mean Modified Bessel function of the first kind 

Mathematics Subject Classification

Primary 26E60 Secondary 26D07 33C10 30E20 33C75 

Notes

Acknowledgements

The authors thank the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Institute of MathematicsHenan Polytechnic UniversityJiaozuo CityChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliao CityChina
  3. 3.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjin CityChina
  4. 4.Customer Service Center, State Grid Zhejiang Electric Power Research InstituteHangzhou CityChina

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