  We study the magnetization process of the spin-1/2 antiferromagnetic
Heisenberg model on a layered triangular lattice by means of a numerical
cluster mean-field method with a scaling scheme (CMF+S). It has been known that
antiferromagnetic spins on a two-dimensional (2D) triangular lattice with
quantum fluctuations exhibit a one-third magnetization plateau in the
magnetization curve under magnetic field. We demonstrate that the CMF+S
quantitatively reproduces the magnetization curve including the stabilization
of the plateau. {We also discuss the effects of a finite interlayer coupling,
which is unavoidable in real quasi-2D materials. It has been recently argued
for a model of the layered-triangular-lattice compound Ba3CoSb2O9 that such
interlayer coupling can induce an additional first-order transition at a strong
field. We present the detailed CMF+S results for the magnetization and
susceptibility curves of the fundamental Heisenberg Hamiltonian in the presence
of magnetic field and weak antiferromagnetic interlayer coupling. The extra
first-order transition appears as a quite small jump in the magnetization curve
and a divergence in the susceptibility at a strong magnetic field ~ 0.712 of
the saturation field.
