Caratheodory inequality for analytic operator function

Suppose H is a complex Hilbert space, A H (Δ) denotes the set of all analytic operator function on Δ, and the set N H (Δ)={f(z)/f(z) is an analytic operator function on the open unit disk Δ, f(z)f(w)=f(w)f(z), f * (z)f(z)=f(z)f * (z), ∀ z, w ≡ δ }. The note proves that if f(z)∈N H (Δ), (or A H (Δ)| f(z)|⩽1, ∀ z∈Δ then |f′(T)|⩽(1-|T| 2 ) −1 |I−f * (T)f(T)| 1/2 |I−f(T)f * (T)| 1/2 , where T∃−(H) (or T * T=TT *, respectively), |T|<1, Tf=fT.