Let
D
D
be an open set in the
d
d
-dimensional Euclidean space
R
d
{{\mathbf {R}}^d}
containing the origin
0
0
and let
h
(
p
)
(
x
,
D
)
{h^{(p)}}(x,D)
be the least harmonic majorant of
|
x
|
p
|x{|^p}
in
D
D
, where
0
>
p
>
∞
0 > p > \infty
if
d
⩾
2
d \geqslant 2
and
1
⩽
p
>
∞
1 \leqslant p > \infty
if
d
=
1
d = 1
. We shall be concerned with the following isoperimetric inequalities
h
(
p
)
(
0
,
D
)
1
/
p
⩽
c
r
(
D
)
{h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)
, where
r
(
D
)
r(D)
denotes the volume radius of
D
D
, namely, a ball with radius
r
(
D
)
r(D)
has the same volume as
D
D
has and
c
c
is a constant dependent on
d
d
and
p
p
but independent of
D
D
. We fix
d
d
and denote by
c
(
p
)
c(p)
the infimum of such constants
c
c
. As a function of
p
p
,
c
(
p
)
c(p)
is nondecreasing and satisfies
c
(
p
)
⩾
1
c(p) \geqslant 1
. We shall show (1) there are positive constants
C
1
{C_1}
and
C
2
{C_2}
such that
C
1
p
(
d
−
1
)
/
d
⩽
c
(
p
)
⩽
C
2
p
(
d
−
1
)
/
d
{C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}
for
p
⩾
1
p \geqslant 1
, (2)
c
(
p
)
=
1
c(p) = 1
if
p
⩽
d
+
2
1
−
d
p \leqslant d + {2^{1 - d}}
. Many estimations of
h
(
p
)
(
0
,
D
)
{h^{(p)}}(0,D)
and their applications are also given.