A New Type of Operator Convexity

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.

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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

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.

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Correspondence to Bich-Khue T. Vo.

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Dinh, T., Dinh, T. & Vo, B.T. A New Type of Operator Convexity. Acta Math Vietnam 43, 595–605 (2018). https://doi.org/10.1007/s40306-018-0259-y

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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