Acta Mathematica Vietnamica
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A New Type of Operator Convexity
Trung-Hoa Dinh
,
Thanh-Duc Dinh
,
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 $$f \left( \left[\lambda A^{r} + (1-\lambda)B^{r}\right]^{1/r}\right) \leq \left[\lambda f(A)^{s} +(1-\lambda)f(B)^{s}\right]^{1/s},$$ 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.