An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space

We prove a sharp isoperimetric inequality for the harmonic mean of the first $m-1$ nonzero Neumann eigenvalues for bounded Lipschitz domains symmetric about the origin in Gauss space. Our result generalizes the Szegö-Weinberger type inequality in Gauss space, as proved in [8, Theorem 4.1].