engleski

Inequalities involving operator superquadratic functions

In this paper, related to the well-known operator convex functions, we study a class of operator functions, the operator superquadratic functions. We present some Jensen-type operator inequalities for these functions. In particular, we show that $f:[0, \infty)\to\mathbb{; ; R}; ; $ is an operator midpoint superquadratic function if and only if $ f\left(C^*AC\right)\leq C^*f(A)C- f\left(\sqrt{; ; C^*A^2C-(C^*AC)^2}; ; \right)$ holds for every positive operator $A\in\mathcal{; ; B}; ; (\mathcal{; ; H}; ; )^+$ and every contraction $C$. As an application, some inequalities for quasi- arithmetic operator means are given.

operator inequality ; operator superquadratic function ; operator convex function ; Jensen operator inequality ; quasi-arithmetic operator mean

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