  We construct and study a generalized one-parameter class of scheme
transformations, denoted $S_{R,m,k_1}$ with $m \ge 2$, with the property that
an $S_{R,m,k_1}$ scheme transformation eliminates the $\ell$-loop terms in the
beta function of a gauge theory from loop order $\ell=3$ to order $\ell=m+1$,
inclusive. These scheme transformations are applied to the higher-loop
calculation of the infrared zero of the beta function of an asymptotically free
gauge theory with multiple fermions. We show that scheme transformations in
this generalized class satisfy a set of criteria for physical acceptability
over a larger range of numbers of fermions than previously studied scheme
transformations. We also present an interesting modification of a different
type of scheme transformation that removes the three-loop term in the beta
function.
