  Constraints on the Higgs boson self-coupling are set by combining
double-Higgs boson analyses in the $b\bar{b}b\bar{b}$, $b\bar{b}\tau^+\tau^-$
and $b\bar{b} \gamma \gamma$ decay channels with single-Higgs boson analyses
targeting the $\gamma \gamma$, $ZZ^*$, $WW^*$, $\tau^+ \tau^-$ and $b\bar{b}$
decay channels. The data used in these analyses were recorded by the ATLAS
detector at the LHC in proton$-$proton collisions at $\sqrt{s}=13$ TeV and
correspond to an integrated luminosity of 126$-$139 fb$^{-1}$. The combination
of the double-Higgs analyses sets an upper limit of $\mu_{HH} < 2.4$ at 95%
confidence level on the double-Higgs production cross-section normalised to its
Standard Model prediction. Combining the single-Higgs and double-Higgs
analyses, with the assumption that new physics affects only the Higgs boson
self-coupling ($\lambda_{HHH}$), values outside the interval $-0.4<
\kappa_{\lambda}=(\lambda_{HHH}/\lambda_{HHH}^{\textrm{SM}})< 6.3$ are excluded
at 95% confidence level. The combined single-Higgs and double-Higgs analyses
provide results with fewer assumptions, by adding in the fit more coupling
modifiers introduced to account for the Higgs boson interactions with the other
Standard Model particles. In this relaxed scenario, the constraint becomes
$-1.4 < \kappa_{\lambda} < 6.1$ at 95% CL.
