Let
M
n
M^n
be an entire graph in the Euclidean
(
n
+
1
)
(n+1)
-space
R
n
+
1
\mathbb R^{n+1}
. Denote by
H
H
,
R
R
and
|
A
|
|A|
, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of
M
n
M^n
. We prove that if the mean curvature
H
H
of
M
n
M^n
is bounded, then
inf
M
|
R
|
=
0
\inf _M|R|=0
, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of
M
n
M^n
is negative, then
inf
M
|
A
|
=
0
\inf _M|A|=0
. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate
inf
|
H
|
,
inf
|
R
|
\inf |H|,\;\inf |R|
and
inf
|
A
|
\inf |A|
for graphs in
R
n
+
1
\mathbb R^{n+1}
of
C
2
C^2
real-valued functions defined on closed balls in
R
n
\mathbb R^n
.