  Let $M$ be a closed spin manifold of dimension $m\geq6$ equipped with a
Riemannian metric $\ig$ and a spin structure $\sa$. Let $\lm_1^+(\tilde\ig)$ be
the smallest positive eigenvalue of the Dirac operator $D_{\tilde\ig}$ on $M$
with respect to a metric $\tilde\ig$ conformal to $\ig$. The B\"ar-Hijazi-Lott
invariant is defined by
$\lm_{min}^+(M,\ig,\sa)=\inf_{\tilde\ig\in[\ig]}\lm_1^+(\tilde\ig)\Vol(M,\tilde\ig)^\frac{1}{m}$.
In this paper, we show that \[
\lm_{min}^+(M,\ig,\sa)<\lm_{min}^+(S^m,\ig_{S^m},\sa_{S^m})=\frac
m2\Vol(S^m,\ig_{S^m})^{\frac1m} \] provided that $\ig$ is not locally
conformally flat. This estimate is a spinorial analogue to an estimate by T.
Aubin, solving the Yamabe problem in this setting.
