  Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that
$\varphi$ is a plurisubharmonic (psh) function on $X$ which is dominated by a
continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show
that $\varphi$ admits a psh extension to $M$ which is dominated by $cu$ on $M$.
  We use this result to prove that any $\omega$-psh function on a subvariety of
the complex projective space is the restriction of a global $\omega$-psh
function, where $\omega$ is the Fubini-Study K\"ahler form.
