Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in C(+) having the property f(z) not equal 0 for 0 < Im z < 1 and we obtain lower bounds for vertical bar f(z)vertical bar for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for z is an element of C(+) and I + A(z) is invertible for 0 < Im z < 1 and is unitary for z is an element of R.