The Parameter Planes of λz m exp ( z ) for m ≥ 2

We consider the families of entire transcendental maps given by Fλ,m(z) = λz m exp(z) where m ≥ 2. All functions Fλ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the inmediate basin, A(0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A(0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of C which serve as a model for Fλ,m, in the sense that they are related by means of quasiconformal surgery to Fλ,m.