Plünnecke proved that if
B
⊆
N
B\subseteq \mathbb {N}
is a basis of order
h
>
1
h>1
, i.e.,
σ
(
h
B
)
=
1
\sigma (hB)=1
, then
σ
(
A
+
B
)
⩾
σ
(
A
)
1
−
1
h
\sigma (A+B)\geqslant \sigma (A)^{1-\frac {1}{h}}
, where
A
A
is an arbitrary subset of
N
\mathbb {N}
and
σ
\sigma
represents Shnirel’man density. In this paper we explore whether
σ
\sigma
can be replaced by other asymptotic densities. We show that Plünnecke’s inequality above is true if
σ
\sigma
is replaced by lower asymptotic density
d
_
\underline {d}
or by upper Banach density
B
D
BD
but not by upper asymptotic density
d
¯
\overline {d}
. The result about
d
_
\underline {d}
has some interesting consequences such as the inequality
d
_
(
A
+
P
)
⩾
d
_
(
A
)
2
/
3
\underline {d}(A+P)\geqslant \underline {d}(A)^{2/3}
for any
A
⊆
N
A\subseteq \mathbb {N}
, where
P
P
is the set of all prime numbers, and the inequality
d
_
(
A
+
C
)
⩾
d
_
(
A
)
3
/
4
\underline {d}(A+C)\geqslant \underline {d}(A)^{3/4}
for any
A
⊆
N
A\subseteq \mathbb {N}
, where
C
C
is the set of all cubes of nonnegative integers. The result about
B
D
BD
generalizes Theorem 3 of a 2001 work of the author by reducing the requirement of
B
B
being a piecewise basis to the requirement of
B
B
being an upper Banach basis.