  We present a new method to determine the best constant of the Sobolev-type
embedding in one dimension with a norm including a bounded inhomogeneous
potential term. This problem is closely connected to the Green function of the
Schr\"odinger operator with inhomogeneous potential. A minimization problem of
a Rayleigh-type quotient in a Sobolev space gives the best constant of the
Sobolev embedding. We decompose the minimization problem into two
sub-minimization problems and show that the Green function provides the
minimizer of the first minimization problem. Then, it enables us to derive a
new precise estimate of the best constant and function for inhomogeneous
bounded potential cases. As applications, we give some examples of the
inhomogeneous potential whose best constant and function of the Sobolev-type
embedding are explicitly determined.
