A logarithmic lower bound for the second Bohr radius

Abstract

The purpose of this note is to obtain an improved lower bound for the multidimensional Bohr radius introduced by L. Aizenberg (2000, Proceedings of the American Mathematical Society 128, 1147–1155), by means of a rather simple argument.

Bohr’s theorem [4] 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 [3] 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 [3, 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 [5] 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 [1] 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 [1] 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 [2], 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 [6], 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 [2, 5] 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 [6, p. 56] (see also Lemma 2.1 of [6]) 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 [2, 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.

Remark 0.2 It should be mentioned that the logarithmic term in the known upper bound for $B(B_{\ell _p^n})$ in (0.3) differs from the logarithmic term in the lower bound for $B(B_{\ell _p^n})$ obtained in Theorem 0.1. Hence, it remains unknown whether this lower bound is asymptotically optimal. Let us also note that for $p=1$ , (0.3) asserts that $B(B_{\ell _1^n})$ is bounded above by $(4\log n)/n$ , but from (0.2) it is clear that the $4\log n$ term can be replaced by a constant less than $1$ . Therefore, it seems that there is room for improvement on the upper bound of $B(B_{\ell _p^n})$ in (0.3) as well, at least for a certain range of p.