We consider Gauss sums of the form
\[
G
n
(
a
)
=
∑
x
∈
F
p
m
χ
(
x
n
)
G_n(a) = \sum _{x \in \mathbb {F}_{p^m}} \chi (x^n)
\]
with a nontrivial additive character
χ
≠
χ
0
\chi \ne \chi _0
of a finite field
F
p
m
\mathbb {F}_{p^m}
of
p
m
p^m
elements of characteristic
p
p
. The classical bound
|
G
n
(
a
)
|
≤
(
n
−
1
)
p
m
/
2
|G_n(a)| \le (n-1) p^{m/2}
becomes trivial for
n
≥
p
m
/
2
+
1
n \ge p^{m/2} + 1
. We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on
|
G
n
(
a
)
|
|G_n(a)|
which is nontrivial for the values of
n
n
of order up to
p
m
/
2
+
1
/
6
p^{m/2 + 1/6}
. We also show that for almost all primes one can obtain a bound which is nontrivial for the values of
n
n
of order up to
p
m
/
2
+
1
/
2
p^{m/2 + 1/2}
.