  An isoperimetric inequality for lower order nonzero Neumann eigenvalues of
the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic
space has been proven in this paper. About this conclusion, we would like to
point out two things:
  It strengthens the well-known Szeg\H{o}-Weinberger inequality for nonzero
Neumann eigenvalues of the classical free membrane problem given in [J.
Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956)
633-636];
  Recently, Xia-Wang [Math. Ann. 385 (2023) 863-879] gave a very important
progress to the celebrated conjecture of M. S. Ashbaugh and R. D. Benguria
proposed in [SIAM J. Math. Anal. 24 (1993) 557-570]. It is easy to see that our
conclusion here covers Xia-Wang's this progress as a special case.
  In this paper, we have also proposed two open problems which can be seen as a
generalization of Ashbaugh-Benguria's conjecture mentioned above.
