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Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 595–605 | Cite as

A New Type of Operator Convexity

  • Trung-Hoa Dinh
  • Thanh-Duc Dinh
  • Bich-Khue T. Vo
Article
  • 81 Downloads

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.

Funding Information

Research of the first and the second authors is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 101.02-2017.310.

References

  1. 1.
    Ando, T., Hiai, F.: Operator log-convex functions and operator means. Math. Ann. 350(3), 611–630 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Audenaert, K.M.R., Hiai, F.: On matrix inequalities between the power means: counterexamples. Linear Algebra Appl. 439(5), 1590–1604 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dinh, T.-H, Vo, B.-K.T.: Some inequalities for operator \((p, h)\)-convex functions. Linear Multilinear Algebra 66(3), 580–592 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gill, P.M., Pearce, C.E.M., Pečarić, J.: Hadamard’s inequality for r-convex functions. J. Math. Anal. Appl. 215(2), 461–470 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Neumark, M.A.: On a representation of additive operator set functions. Dok. Akad. Nauk SSSR 41(9), 373–375 (1943). (Russian); English translation: C. R. (Doklady) Akad. Sci. URSS (N.S.) 41, 359–361 (1943)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Tikhonov, O.E.: A note on definition of matrix convex functions. Linear Algebra Appl. 416(2–3), 773–775 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhang, K.S., Wan, J.P.: p-convex functions and their properties. Pure Appl. Math. (Xi’an) 23(1), 130–133 (2007)MathSciNetzbMATHGoogle Scholar

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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