Let
K
(
B
ℓ
p
n
,
B
ℓ
q
n
)
K(B_{\ell _p^n},B_{\ell _q^n})
be the
n
n
-dimensional
(
p
,
q
)
(p,q)
-Bohr radius for holomorphic functions on
C
n
\mathbb {C}^n
. That is,
K
(
B
ℓ
p
n
,
B
ℓ
q
n
)
K(B_{\ell _p^n},B_{\ell _q^n})
denotes the greatest number
r
≥
0
r\geq 0
such that for every entire function
f
(
z
)
=
∑
α
a
α
z
α
f(z)=\sum _{\alpha } a_{\alpha } z^{\alpha }
in
n
n
-complex variables, we have the following (mixed) Bohr-type inequality:
sup
z
∈
r
⋅
B
ℓ
q
n
∑
α
|
a
α
z
α
|
≤
sup
z
∈
B
ℓ
p
n
|
f
(
z
)
|
,
\begin{equation*} \sup _{z \in r \cdot B_{\ell _q^n}} \sum _{\alpha } \vert a_{\alpha } z^{\alpha } \vert \leq \sup _{z \in B_{\ell _p^n}} \vert f(z) \vert , \end{equation*}
where
B
ℓ
r
n
B_{\ell _r^n}
denotes the closed unit ball of the
n
n
-dimensional sequence space
ℓ
r
n
\ell _r^n
.
For every
1
≤
p
,
q
≤
∞
1 \leq p, q \leq \infty
, we exhibit the exact asymptotic growth of the
(
p
,
q
)
(p,q)
-Bohr radius as
n
n
(the number of variables) goes to infinity.