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 Gunthers inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.
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