The Hessian quotient equations
S
k
,
l
(
D
2
u
)
≡
S
k
(
D
2
u
)
S
l
(
D
2
u
)
=
1
,
∀
x
∈
R
n
\begin{equation} S_{k,l}(D^2u)\equiv \frac {S_k(D^2u)}{S_l(D^2u)}=1, \ \ \forall x\in {\mathbb {R}}^n \end{equation}
were studied for
k
−
k-
th symmetric elementary function
S
k
(
D
2
u
)
S_k(D^2u)
of eigenvalues
λ
(
D
2
u
)
\lambda (D^2u)
of the Hessian matrix
D
2
u
D^2u
, where
0
≤
l
>
k
≤
n
0\leq l>k\leq n
. For
l
=
0
l=0
, (0.1) is reduced to a
k
−
k-
Hessian equation
S
k
(
D
2
u
)
=
1
,
∀
x
∈
R
n
.
\begin{equation} S_k(D^2u)=1, \ \ \forall x\in {\mathbb {R}}^n. \end{equation}
Two quadratic growth conditions were found by Bao-Cheng-Guan-Ji [American J. Math. 125 (2013), pp. 301–316] ensuring the Bernstein properties of (0.1) and (0.2) respectively. In this paper, we will drop the point wise quadratic growth condition of Bao-Cheng-Guan-Ji 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
L
p
L^p
-integrability respectively. Our new volume growth or
L
p
−
L^p-
integrable conditions improve largely various previously known point wise conditions provided Bao et al.; Chen and Xiang [J. Differential Equations 267 (2019), pp. 52027–5219]; Cheng and Yau [Comm. Pure Appl. Math. 39 (1986), pp. 8397–866]; Li, Ren, and Wang [J. Funct. Anal. 270 (2016), pp. 26917–2714]; Yuan [Invent. Math. 150 (2002), pp. 1177–125], etc.