Gradient inequalities for an integral transform of positive operators in Hilbert spaces - Dragomir, Silvestru Sever (Annales Mathematicae Silesianae, 2023)
For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0,∞) we consider the following integral transform D(w,μ)(T) := ∫0∞w(λ)(λ+T)-1dμ(λ), where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. Assume that A≥α>0, δ≥B>0 and 0<m≤B-A≤M for some constants α, δ, m, M. Then 0 ≤ -mD'(w,μ)(δ) ≤ D(w,μ)(A)-D(w,μ)(B) ≤ -MD'(w,μ)(α), where D'(w,μ)(t) is the derivative of D(w,μ)(t) as a function of t>0. If f:[0,∞)→ℝ is operator monotone on [0,∞) with f(0)=0, then 0 ≤ m/δ2[f(δ)-f'(δ)δ] ≤ f(A)A-1-f(B)B-1 ≤ M/α2[f(α)-f'(α)α]. Some examples for operator convex functions as well as for integral transforms D(·,·) related to the exponential and logarithmic functions are also provided.