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.
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.