Abstract: In this paper,it is proved that if A is a normal proper contraction on Hilbert space H and F(z)=U(z)+iV(z) is operator-valued analytic on the unit disc △ and 0 < p < 1,then||F(A)||^p ≤ ||F(0)||^p+C_p(1-||A||)^-2× sup_{0 < t < 1}∬_△||U[(At-εI)(I-εA·t)^-1]||^pdm(ε).

Key words: Hilbert space, operator, operator-valued function, proper contraction