  We present analytical results for the leading top-Yukawa and QCD contribution
to the $\beta$-function for the Higgs self-coupling $\lambda$ of the Standard
Model at four-loop level, namely the part $\propto y_t^4 g_s^6$ independently
confirming a result given in [1]. We also give the contribution $\propto y_t^2
g_s^6$ of the anomalous dimension of the Higgs field as well as the terms
$\propto y_t g_s^8$ to the top-Yukawa $\beta$-function which can also be
derived from the anomalous dimension of the top quark mass. We compare the
results with the RG functions of the correlators of two and four scalar
currents in pure QCD and find a new relation between the anomalous dimension
$\gamma_0$ of the QCD vacuum energy and the anomalous dimension $\gamma_m^{SS}$
appearing in the RG equation of the correlator of two scalar currents. Together
with the recently computed top-Yukawa and QCD contributions to $\beta_{g_s}$
[2,3] the $\beta$-functions presented here constitute the leading four-loop
contributions to the evolution of the Higgs self-coupling. A numerical estimate
of these terms at the scale of the top-quark mass is presented as well as an
analysis of the impact on the evolution of $\lambda$ up to the Planck scale and
the vacuum stability problem.
