  We analyze an upper bound on the curvature of a Riemannian manifold, using
"root-Ricci" curvature, which is in between a sectional curvature bound and a
Ricci curvature bound. (A special case of root-Ricci curvature was previously
discovered by Osserman and Sarnak for a different but related purpose.) We
prove that our root-Ricci bound implies G\"unther's inequality on the candle
function of a manifold, thus bringing that inequality closer in form to the
complementary inequality due to Bishop.
