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In the Hyers-Ulam-Rassias stability depending on the type of the function whose stability we want to verify a suitable functional equation is used. The authors in [6] want to investigate the Hyers-Ulam-Rassias stability of n-Jordan C∗ - homomorphisms on C∗ -algebras, but they used a quadratic functional equation while we know that the homomorphisms are linear on C∗ -algebras. In this paper, we correct the main results of [6] by removing the quadratic functional equation and replacing the linear one and removing some extra conditions. We also show that by using some other multi-variable linear functional equations, the estimation becomes better and more accurate. [ABSTRACT FROM AUTHOR]

Lieb’s extension theorem holds for generalized p + q ∈ [0; 1] and Ando convexity theorem holds for q - r > 1. In this paper, we give a complete characterization for concavity or convexity of Lieb well known theorem in the case where p + q ≥ 1 or p+q ≤ 0. We also characterize some auxiliary results including Ando theorem for q-r ≤ 1.

In this paper, we give a new characterization for the perspective of a continuous function under certain assumptions. This result generalizes a non-commutative analogue of the arithmetic–geometric mean inequality. Our main result extends and strengthens the existing result concerning the characterization of the geometric operator mean. We provide the characterization of other operator means, in particular, the harmonic operator mean and the arithmetic operator mean. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the Tsallis relative operator φ-entropy between two strictly positive operators and verify its properties such as joint convexity, joint sub additivity, and monotonicity. We also give the Shannon type operator inequality and its reverse one satisfied by the Tsallis relative operator φ-entropy. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the lower and upper bound of the relative operator (α , β) -entropy. We refine and improve the lower and upper bound of the relative operator entropy and generalized relative operator entropy. As a consequence of our result, the bounds of the relative operator entropy announced by Fujii and Kamei will improve. [ABSTRACT FROM AUTHOR]