Bohr’s theorem [Reference Bohr4] states that for each bounded holomorphic self-mapping $f(z)=\sum _{k=0}^\infty a_kz^k$ of the open unit disk $\mathbb {D}$ , we have

$$ \begin{align*}\sum_{k=0}^\infty|a_k|\left(\frac{1}{3}\right)^k\leq 1, \end{align*} $$

and this quantity $1/3$ is the best possible. In an attempt to generalize this result in higher dimensions, the first Bohr radius $K(R)$ for a bounded complete Reinhardt domain $R\subset \mathbb {C}^n$ was defined in [Reference Boas and Khavinson3] by Boas and Khavinson. Namely, $K(R)$ is the supremum of all $r\in [0, 1]$ such that for each holomorphic function $f(z)=\sum _{\alpha }a_\alpha z^\alpha $ on R with $|f(z)|\leq 1$ for all $z\in R$ , we have

$$ \begin{align*}\sum_{\alpha}\left|a_\alpha z^\alpha\right|\leq 1 \end{align*} $$

for all $z\in rR$ . Let us clarify here that a complete Reinhardt domain R in $\mathbb {C}^n$ is a domain such that if $z=(z_1, z_2, \ldots , z_n)\in R$ , then $(\lambda _1z_1, \lambda _2z_2, \ldots , \lambda _nz_n)\in R$ for all $\lambda _i\in \overline {\mathbb {D}}, 1\leq i\leq n$ . Of particular interest to us are the Reinhardt domains

$$ \begin{align*} B_{\ell_p^n}:=\{z\in\mathbb{C}^n: \|z\|_p<1\}, \end{align*} $$

where $\ell _p^n$ is the Banach space $\mathbb {C}^n$ equipped with the p-norm $\|z\|_p:=\left (\sum _{k=1}^n|z_k|^p\right )^{1/p}$ for $1\leq p<\infty $ , while $\|z\|_\infty :=\max _{1\leq k\leq n}|z_k|$ . Also, we use the standard multi-index notation: $\alpha $ denotes an n-tuple $(\alpha _1, \alpha _2,\ldots , \alpha _n)$ of nonnegative integers, $|\alpha |:=\alpha _1+\alpha _2+\cdots +\alpha _n$ , and for $z=(z_1, z_2, \ldots , z_n)\in \mathbb {C}^n$ , $z^\alpha $ is the product $z_1^{\alpha _1}z_2^{\alpha _2}\cdots z_n^{\alpha _n}$ . Indeed, $K(\mathbb {D})=1/3$ , and it is known from [Reference Boas and Khavinson3, Theorem 3] that $K(R)\geq K(B_{\ell _\infty ^n})$ for any complete Reinhardt domain R. Through the substantial progress made in a series of papers from 1997 to 2011, it was finally concluded by Defant and Frerick in [Reference Defant and Frerick5] that there exists a constant $c\geq 0$ such that for each $p\in [1, \infty ]$ ,

(0.1) $$ \begin{align} \frac{1}{c}\left(\frac{\log n}{n}\right)^{1-\frac{1}{\min\{p, 2\}}} \leq K(B_{\ell_p^n})\leq c\left(\frac{\log n}{n}\right)^{1-\frac{1}{\min\{p, 2\}}} \end{align} $$

for all $n>1$ .

On the other hand, Aizenberg [Reference Aizenberg1] introduced a second Bohr radius $B(R)$ for a bounded complete Reinhardt domain $R\subset \mathbb {C}^n$ , which is the largest $r\in [0, 1]$ such that for each holomorphic function $f(z)=\sum _{\alpha }a_\alpha z^\alpha $ on R satisfying $|f(z)|\leq 1$ for all $z\in R$ , we have

$$ \begin{align*} \sum_{\alpha}\sup_{z\in rR}\left|a_\alpha z^\alpha\right|\leq 1. \end{align*} $$

Clearly, $B(\mathbb {D})=1/3$ and $B(B_{\ell _\infty ^n})=K(B_{\ell _\infty ^n})$ . It was also shown in [Reference Aizenberg1] that $B(R)\geq 1-(2/3)^{1/n}>1/(3n)$ for any bounded complete Reinhardt domain $R\subset \mathbb {C}^n (n\geq 2$ ), and that

(0.2) $$ \begin{align} B(B_{\ell_1^n})<\frac{0.446663}{n}. \end{align} $$

Further advances were made by Boas in [Reference Boas2], showing that for all $p\in [1, \infty ]$ ,

(0.3) $$ \begin{align} \frac{1}{3}\left(\frac{1}{n}\right)^{\frac{1}{2}+\frac{1}{\max\{p, 2\}}} \leq B(B_{\ell_p^n})<4\left(\frac{\log n}{n}\right)^{\frac{1}{2}+\frac{1}{\max\{p, 2\}}}\,(n>1). \end{align} $$

To the best of our knowledge, except for the subsequent article [Reference Defant, García and Maestre6], the problem of estimating $B(B_{\ell _p^n})$ has not been considered ever since. This is probably because no specific application of this second Bohr radius seems to be known. However, we believe that this is a problem of independent interest. Our aim is to point out that the results of [Reference Boas2, Reference Defant and Frerick5] readily yield a much refined lower bound for $B(B_{\ell _p^n})$ . This bound shows that analogous to $K(B_{\ell _p^n})$ , $B(B_{\ell _p^n})$ must also contain a $\log n$ term. It may also be noted that for a variety of bounded complete Reinhardt domains $R\subset \mathbb {C}^n$ , parts of our arguments could be adopted to derive results for $B(R)$ from previously known results for $K(R)$ .

To facilitate our discussion, let us now denote by $\chi _{\mathrm{mon}}(\mathcal {P}({}^{m}\ell _p^n))$ the unconditional basis constant associated with the basis consisting of the monomials $z^\alpha $ , for the space $\mathcal {P}({}^{m}\ell _p^n)$ of m-homogeneous complex-valued polynomials P on $\ell _p^n$ . This space is equipped with the norm $\|P\|=\sup _{\|z\|_p\leq 1}|P(z)|$ . We mention here that a Schauder basis $(x_n)$ of a Banach space X is said to be unconditional if there exists a constant $c\geq 0$ such that

$$ \begin{align*} \left\|\sum_{k=1}^t\epsilon_k\alpha_k x_k\right\|\leq c\left\|\sum_{k=1}^t\alpha_k x_k\right\| \end{align*} $$

for all $t\in \mathbb {N}$ and for all $\epsilon _k, \alpha _k\in \mathbb {C}$ with $|\epsilon _k|\leq 1$ , $1\leq k\leq t$ . The best constant c is called the unconditional basis constant of $(x_n)$ . Now, it is known from [Reference Defant, García and Maestre6, p. 56] (see also Lemma 2.1 of [Reference Defant, García and Maestre6]) that

(0.4) $$ \begin{align} \chi_{\mathrm{mon}}(\mathcal{P}({}^{m}\ell_p^n)) =\frac{1}{(K_m(B_{\ell_p^n}))^m}, \end{align} $$

where $K_m(B_{\ell _p^n})$ is the supremum of all $r\in [0, 1]$ such that for each m-homogeneous complex-valued polynomial $P(z)=\sum _{|\alpha |=m}a_\alpha z^\alpha $ with $|P(z)|\leq 1$ for all $z\in B_{\ell _p^n}$ , we have $\sum _{|\alpha |=m}\left |a_\alpha z^\alpha \right |\leq 1$ for all $z\in rB_{\ell _p^n}$ . Clearly, $K_m(B_{\ell _p^n})\geq K(B_{\ell _p^n})$ . These facts are instrumental in proving Theorem 0.1.

Theorem 0.1 There exists a constant $C>0$ such that for each $p\in [1, \infty ]$ ,

$$ \begin{align*}B(B_{\ell_p^n})\geq C\frac{(\log n)^{1-\frac{1}{\min\{p, 2\}}}}{n^{\frac{1}{2}+\frac{1}{\max\{p, 2\}}}} \end{align*} $$

for all $n>1$ .

Proof It is observed in [Reference Boas2, p. 335] that

$$ \begin{align*} B(B_{\ell_p^n})\geq \frac{B(B_{\ell_\infty^n})}{n^{\frac{1}{p}}}. \end{align*} $$

Since $B(B_{\ell _\infty ^n})=K(B_{\ell _\infty ^n})\geq C(\sqrt {\log n}/\sqrt {n})$ for some constant $C>0$ (see (0.1)), the proof for the case $p\in [2, \infty ]$ follows immediately from the above inequality.

For the case $p\in [1, 2)$ , a little more work is needed. Given any holomorphic function $f(z)=\sum _{\alpha }a_\alpha z^\alpha $ on $B_{\ell _p^n}$ with $|f(z)|\leq 1$ for all $z\in B_{\ell _p^n}$ , it is evident that for any fixed $z\in B_{\ell _p^n}$ , $h(u):=f(uz)=a_0+\sum _{m=1}^\infty \left (\sum _{|\alpha |=m}a_\alpha z^\alpha \right )u^m:\mathbb {D}\to \overline {\mathbb {D}}$ is a holomorphic function of $u\in \mathbb {D}$ . The well-known Wiener’s inequality asserts that

$$ \begin{align*} \sup_{\|z\|_p\leq 1}\left|\sum_{|\alpha|=m}a_\alpha z^\alpha\right|\leq 1-|a_0|^2 \end{align*} $$

for all $m\geq 1$ . The definition of $\chi _{\mathrm{mon}}(\mathcal {P}({}^{m}\ell _p^n))$ guarantees that for the choices of $\epsilon _\alpha $ s such that $\epsilon _\alpha a_\alpha =|a_\alpha |$ ,

$$ \begin{align*} \left(\sum_{|\alpha|=m}|a_\alpha|\right)\frac{1}{n^{\frac{m}{p}}}=\sum_{|\alpha|=m}|a_\alpha|\left(\frac{1}{n^{\frac{1}{p}}}\right)^\alpha &\leq\sup_{\|z\|_p\leq 1}\left|\sum_{|\alpha|=m}\epsilon_\alpha a_\alpha z^\alpha\right|\\ &\leq \chi_{\mathrm{mon}}(\mathcal{P}({}^{m}\ell_p^n))\sup_{\|z\|_p\leq 1}\left|\sum_{|\alpha|=m}a_\alpha z^\alpha\right|\\ &\leq (1-|a_0|^2)\frac{1}{(K_m(B_{\ell_p^n}))^m} \end{align*} $$

(see (0.4)). That is to say,

$$ \begin{align*} \sum_{|\alpha|=m}|a_\alpha|\leq (1-|a_0|^2)\frac{n^{\frac{m}{p}}}{(K_m(B_{\ell_p^n}))^m}\leq (1-|a_0|^2)\frac{n^{\frac{m}{p}}}{(K(B_{\ell_p^n}))^m}. \end{align*} $$

A little computation reveals that

$$ \begin{align*} \sum_{\alpha}\sup_{z\in rB_{\ell_p^n}}|a_\alpha z^\alpha|&=|a_0|+\sum_{m=1}^\infty r^m\sum_{|\alpha|=m}|a_\alpha|\left(\frac{\alpha^\alpha}{m^m}\right)^{\frac{1}{p}}\\ &\leq |a_0|+\sum_{m=1}^\infty r^m\sum_{|\alpha|=m}|a_\alpha|\\ &\leq|a_0|+(1-|a_0|^2)\sum_{m=1}^\infty\left(\frac{rn^{\frac{1}{p}}}{K(B_{\ell_p^n})}\right)^m. \end{align*} $$

It is clear from the above inequality that $\sum _{\alpha }\sup _{z\in rB_{\ell _p^n}}|a_\alpha z^\alpha |\leq 1$ whenever

$$ \begin{align*}r\leq\frac{1}{3}\left(\frac{K(B_{\ell_p^n})}{n^{\frac{1}{p}}}\right), \end{align*} $$

i.e., $B(B_{\ell _p^n})\geq K(B_{\ell _p^n})/(3n^{1/p})$ . In view of the inequalities (0.1), this completes the proof.