Lipschitz type inequalities for noncommutative perspectives of operator monotone functions in Hilbert spaces - Dragomir, Silvestru Sever (Advances in operator theory, 2021)
Assume that f : [ 0 , ∞ ) → R is a continuous function. We can define the perspective P f B , A by setting P f B , A : = A 1 / 2 f A - 1 / 2 B A - 1 / 2 A 1 / 2 , where A , B > 0 . We show in this paper among others that P f B , P - P f A , P ≤ P 2 B - A p 2 P f m 2 , p - P f m 1 , p m 2 - m 1 if m 1 ≠ m 2 , f ′ m p if m 1 = m 2 = m for all A ≥ m 1 > 0 , B ≥ m 2 > 0 and P ≥ p > 0 . If f is operator monotone on [ 0 , ∞ ) , then for all C ≥ n 1 > 0 , D ≥ n 2 > 0 , Q > q > 0 we also have P f Q , D - P f Q , C ≤ Q 2 D - C q 2 P f q , n 2 - P f q , n 1 n 2 - n 1 if n 2 ≠ n 1 , f q n - q n f ′ q n if n 2 = n 1 = n . Some applications for weighted operator geometric mean and relative operator entropy are also given.