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.