The multiferroic behavior of rare-earth manganites is studied within a microscopic model including a symmetry-allowed magnetoelectric coupling between polarization and magnetization. The magnetic subsystem is described by a frustrated Heisenberg spin model, whereas the ferroelectric subsystem is characterized by an Ising model in a transverse field. Using Green's function method we find analytically the temperature and wave vector dependent elementary excitation of the magnetoelectric system, the polarization and the magnetization for different magnetoelectric coupling strengths. The system undergoes a magnetic transition at T_N and a further reduction of the temperature leads to a ferroelectric transition at T_C < T_N depending on the coupling strength. That coupling is also manifested as a kink in the magnetization and the elementary excitation at T_C. Due to the magnetoelectric coupling the excitation energy exhibits a gap at zero wave vector which increases with increasing coupling. We show that the macroscopic magnetization can be slightly enhanced by an external electric field nearby T_C. An external magnetic field leads to an increase of the polarization.