  In this article, we prove an isoperimetric inequality for the harmonic mean
of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in
$n$-dimensional Hyperbolic space. Our approach to prove this result also gives
a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded
domains in $n$-dimensional Euclidean space.
