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A New Type of Operator Convexity


Trung-Hoa Dinh, Thanh-Duc Dinh, Bich-Khue T. Vo

Source title: 
Acta Mathematica Vietnamica, 43(4): 595-605, 2018 (Scopus)
Academic year of acceptance: 

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

f ([λAr+(1−λ)Br]1/r)≤[λf(A)s+(1−λ) f(B)s]1/s,

where   A, B  are positive definite matrices and for any   λ∈[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.