  By using a point canonical transformation starting from the constant-mass
Schr\"odinger equation for the isotonic potential, it is shown that a
semiconfined harmonic oscillator model with a position-dependent mass in the
BenDaniel-Duke setting and the same spectrum as the standard harmonic
oscillator can be easily constructed and extended to a semiconfined shifted
harmonic oscillator, which could result from the presence of a uniform
gravitational field. A further generalization is proposed by considering a
$m$-dependent position-dependent mass for $0<m<2$ and deriving the associated
semiconfined potential. This results in a family of position-dependent mass and
potential pairs, to which the original pair belongs as it corresponds to $m=1$.
Finally, the potential that would result from a general von Roos kinetic energy
operator is presented and the examples of the Zhu-Kroemer and
Mustafa-Mazharimousavi settings are briefly discussed.
