  We show that there exists a universal positive constant $\varepsilon_0 > 0$
with the following property: Let $g$ be a positive Einstein metric on $S^4$. If
the Yamabe constant of the conformal class $[g]$ satisfies
  $$ Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0 $$
where $g_{\mathbb S}$ denotes the standard round metric on $S^4$, then, up to
rescaling, $g$ is isometric to $g_{\mathbb S}$.
  This is an extension of Gursky's gap theorem for positive Einstein metrics on
the four-sphere.
