  We establish quaternionic contact (qc) versions of the so called Almost Schur
Lemma, which give estimations of the qc scalar curvature on a compact qc
manifold to be a constant in terms of the norm of the $[-1]$-component and the
norm of the trace-free part of the $[3]$-component of the horizontal qc Ricci
tensor and the torsion endomorphism, under certain positivity conditions.
