The relative operator entropy has properties like operator means. In addition, the relative operator entropy has entropy-like properties. In this paper, we prove a Loewner–Heinz type inequality for operator monotone and multiplicative-additive functions and its converse and apply it to the relative operator entropy. We identify the equivalence relations among inequalities for some relative operator entropies in a view point of the perspective.

中文翻译：

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一些相对算子熵不等式之间的等价关系

相对算子熵具有像算子平均值的属性。另外，相对算符熵具有类似熵的性质。在本文中，我们证明了算子单调和乘法加法函数及其逆的Loewner-Heinz型不等式，并将其应用于相对算子熵。我们从透视图的角度确定了一些相对算符熵的不等式之间的等价关系。