  Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We
obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann
eigenvalues and Steklov eigenvalues involving the conformal volume and relative
conformal volume, respectively. We also give an optimal sharp extrinsic upper
bound for closed submanifolds in the space forms. These extend the previous
related results for the first nonzero eigenvalue.
