2020/08/30 by A. G. Ghazanfari, Ghazanfari, Amir Ghasem
Mathematics · #26D10 #26D15 #47A63 #Analytic and geometric function theory #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications
paper · pdf · doi:10.48550/arxiv.2008.13226
openalex publication_date 2020/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a positive definite operator on a Hilbert space H, and |||.||| be a unitarily invariant norm on B(H). We show that if f is an operator monotone function on (0,∞) and n∈ ℕ, then |||Dn f(A)|||≤‖f(n)(A)‖ and ‖f(n)(⋅)‖ is a quasi-convex function on the set of all positive definite operators in B(H). We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-convex or s-convex. As applications, we obtain some of bounds for |||f(B)-f(A)||| in term of |||B-A|||. For instance, Let f,g be two operator monotone functions on (0,∞). Then, for every unitarily invariant norm |||.||| and every positive definite operators A,B, amp;|||f(A)g(A)-f(B)g(B)|||
amp;≤|||B-A|||[max\‖f'(A)‖,‖f'(B)‖\×max\‖g(A)‖,‖g(B)‖\
amp;+max\‖f(A)‖,‖f(B)‖\× max\‖g'(A)‖,‖g'(B)‖\].