Bounds for the derivative of certain meromorphic functions and on meromorphic Bloch-type functions

It is known that if f is holomorphic in the open unit disc \(\mathbb{D}\) of the complex plane and if, for some c > 0, ∣f(z)∣ ⩽ 1/(1−∣z∣²)^c, \(z \in \mathbb{D}\), then ∣f′(z)∣ ⩽ 2(c+1)/(1−∣z∣²)^c+1. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in D. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.