Some inequalities for operator (p, h)-convex functions

Abstract

Let p be a positive number and h a function on satisfying for any . A non-negative continuous function f on is said to be operator (p, h)-convex if

holds for all positive semidefinite matrices A, B of order n with spectra in K, and for any . In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen–Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi–Davis–Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions.

Keywords:

• Operator (p
•  h)-convex functions
• operator Jensen type inequality
• operator Hansen–Pedersen type inequality

AMS Subject Classifications:

• 15B57
• 46L30
• 15A45

Acknowledgements

The authors would like to express sincere thanks to the anonymous referee for his comments which improve this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.04-2014.40]. This work was partially finished at Ton Duc Thang University, Vietnam.