  We show that a very good analytical approximation of Saffman-Delbr\"uck's
(SD) law (mobility of a bio-membrane inclusion) can be obtained easily from the
velocity field produced by a pointlike force in a 2D fluid embedded in a
solvent, by using a small wavelength cutoff of the order of the particle's
radius~$a$. With this method, we obtain analytical generalizations of the SD
law that take into account the bilayer nature of the membrane and the
intermonolayer friction $b$. We also derive, in a calculation that consistently
couples the quasi-planar two-dimensional (2D) membrane flow with the 3D solvent
flow, the correction to the SD law arising when the inclusion creates a local
spontaneous curvature. For an inclusion spanning a flat bilayer, the SD law is
found to hold simply upon replacing the 2D viscosity $\eta_2$ of the membrane
by the sum of the monolayer viscosities, without influence of $b$ as long as
$b$ is above a threshold in practice well below known experimental values. For
an inclusion located in only one of the two monolayers (or adhering to one
monolayer), the SD law is influenced by $b$ when $b<\eta_2/(4a^2)$. In this
case, the mobility can be increased by up to a factor of two, as the opposite
monolayer is not fully dragged by the inclusion. For an inclusion creating a
local spontaneous curvature, we show that the total friction is the sum of the
SD friction and that due to the pull-back created by the membrane deformation,
a point that was assumed without demonstration in the literature.
