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A New Type of Operator Convexity

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

Abstract

Let $r, s$ be positive numbers. We define a new class of operator $(r, s)$ -convex functions by the following inequality $f ({[λ A^{r} + (1 - λ) B^{r}]}^{1 / r}) \leq {[λ f (A)^{s} + (1 - λ) f (B)^{s}]}^{1 / s},$ where $A, B$ are positive definite matrices and for any $λ \in [0, 1]$ . 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.

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A New Type of Operator Convexity

Abstract