  The Hessian quotient equatio were studied for k-th symmetric elementary
function S_k(D^2u) of eigenvalues of the Hessian matrix D^2u. Two pointwise
quadratic growth conditions were found by Bao-Cheng-Guan-Ji ([1], American J.
Math., 2003, 125, 301-316) ensuring Bernstein properties of Hessian quotient
equation or k-Hessian equation respectively. In this paper, we will drop the
point wise quadratic growth condition of [1] and prove three necessary and
sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse
isoperimetric type inequality, volume growth or Lp-integrable respectively.Our
volume growth or Lp-integrable conditions improve largely various known point
wise conditions in [1,6,7,13,18] etc.
