  In this paper we introduce a new technique for proving norm inequalities in
operator ideals with an unitarily invariant norm. Among the well known
inequalities which can be proved with this technique are the L\"owner-Heinz
inequality, inequalities relating various operator means and the
Corach-Porta-Recht inequality. We prove two general inequalities and from them
we derive several inequalities by specialization, many of them new. We also
show how some inequalities, known to be valid for matrices or bounded
operators, can be extended with this technique to normed ideals in
$C^*$-algebras, in particular to the noncommutative $L^p$-spaces of a
semi-finite von Neumann algebra.
