A New Type of Operator Convexity

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

f ([λA^r+(1−λ)B^r]^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.