In this paper, we introduce two notions of a relative operator (α, β)-entropy and a Tsallis relative operator (α, β)-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning α and β. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.Significance Statement. What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics. Our proof strategy shows that the joint convexity of the perspective of some functions plays a crucial role to give simple proofs for the joint convexity (resp. concavity) of some relative operator entropies.

中文翻译：

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具有透视方法的一些相对算子熵的参数扩展的凸性

在本文中，我们介绍了相对算子的两个概念（α,β)-熵和 Tsallis 相对算子 (α,β)-熵作为相对算子熵和 Tsallis 相对算子熵的两个参数扩展。我们应用透视方法来证明这些新概念的联合凸性或凹性，在某些条件下，α和β. 事实上，我们给出了参数扩展，但是以这样一种方式，它们保持共同凸或共同凹。意义声明。这里的新颖之处在于，我们令人信服地证明了我们的技术如何用于为与量子统计相关的函数的新旧定理提供简单的证明。我们的证明策略表明，某些函数视角的联合凸性对于给出一些相对算子熵的联合凸性（分别是凹性）的简单证明起着至关重要的作用。