We propose and numerically investigate annular-cladding erbium doped multicore fibers (AC-EDMCF) with either solid or air hole inner cladding to enhance the pump power efficiency in optical amplifiers for spatial division multiplexing (SDM) transmission links. We first propose an all-glass fiber in which a central inner cladding region with a depressed refractive index is introduced to confine the pump inside a ring-shaped region overlapping the multiple signal cores. Through numerical simulations, we determine signal core and annular pump cladding parameters respecting fabrication constraints. We also propose and examine a multi-spot injection scheme for launching the pump in the annular cladding. With this all-glass fiber with annular cladding, our results predict 10 dB increase in gain and 21% pump power savings compared to the standard double cladding design. We also investigate a fiber with an air hole inner cladding to further enhance the pump power confinement and minimize power leaking into the inner cladding. The results are compared to the all-glass AC-EDMCF.