A New Type of Operator Convexity

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Abstract

Let \(r, s\) be positive numbers. We define a new class of operator \((r, s)\)-convex functions by the following inequality

$$ f \left( \left[\lambda A^{r} + (1-\lambda)B^{r}\right]^{1/r}\right) \leq \left[\lambda f(A)^{s} +(1-\lambda)f(B)^{s}\right]^{1/s}, $$
where \(A, B\) are positive definite matrices and for any \(\lambda \in [0,1]\). We prove the Jensen, Hansen-Pedersen, and Rado type inequalities for such functions. Some equivalent conditions for a function f to become operator \((r, s)\)-convex are established.

Keywords

Operator \({(r, \protect s)}\)-convex functions Operator Jensen type inequality Operator Hansen-Pedersen type inequality Operator Rado type inequality 

Mathematics Subject Classification (2010)

46L30 15A45 15B57 

Notes

Acknowledgements

The authors would like to thank Professor Fumio Hiai and the referee for useful comments which improved the quality of the present paper.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Trung-Hoa Dinh
    • 1
    • 2
    • 3
  • Thanh-Duc Dinh
    • 4
  • Bich-Khue T. Vo
    • 4
    • 5
  1. 1.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Civil EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Department of Mathematics and StatisticsUniversity of North FloridaJacksonvilleUSA
  4. 4.Quy Nhon UniversityQuy NhonVietnam
  5. 5.University of Finance - MarketingHo Chi Minh CityVietnam

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