Bohr-Rogosinski in Several Complex Variables

• Bohr-Rogosinski phenomenon is defined by the quantitative relation between the modulus of bounded holomorphic functions on the unit polydisc and the sum of their Taylor coefficients.
• The methodology employs sharp radius determination via explicit root equations and extremal functions, extending classical univariate results to multivariate and Banach space contexts.
• Applications include derivative and area-type refinements that enhance classical inequalities, impacting studies on vector-valued, lacunary, and multivariate holomorphic functions.

The Bohr-Rogosinski phenomenon in several complex variables concerns the quantitative relationship between the modulus of a bounded holomorphic function on the unit polydisc and the sum of the moduli of its Taylor coefficients, or generalizations thereof, with pointwise or functionally refined terms. This theory seeks sharp radii ("Bohr-Rogosinski radii") up to which inequalities—analogous to the classical univariate Bohr and Rogosinski inequalities—hold in the multivariable setting, extending also to vector-valued, lacunary, and Banach-space contexts. Recent works have provided definitive sharp constants, established multidimensional analogues of univariate results, and incorporated directional (Euler operator) growth and area-based improvements (Ahamed et al., 10 Jan 2026, Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).

1. Framework: Domains, Function Classes, and Notation

The unit polydisc in $\mathbb{C}^n$ is defined as

$$\mathbb{D}^n = \{\, z=(z_1,\dots,z_n)\in\mathbb{C}^n : |z_j|<1,\; j=1,\dots,n\,\}$$

with the sup-norm $\|z\|_\infty = \max_j{|z_j|}$. A multi-index $\alpha=(\alpha_1,...,\alpha_n)\in\mathbb{N}^n$ has length $|\alpha| = \alpha_1+\cdots+\alpha_n$ and monomial $z^\alpha = z_1^{\alpha_1}\cdots z_n^{\alpha_n}$. Holomorphic functions $f:\mathbb{D}^n\to\mathbb{C}$ admit power series expansions

$f(z) = \sum_{|\alpha|\ge 0} a_\alpha z^\alpha$

where the convergence holds for $z\in \mathbb{D}^n$. The class $\mathcal{B}_{n,m}$ denotes multivariate Schwarz functions vanishing to order $m$ at the origin: $$\mathcal{B}_{n,m} = \left\{ \omega(z) = (\omega_1(z_1), \dots, \omega_n(z_n)) : \omega_j^{(k)}(0) = 0 \text{ for } 0\leq k < m, \ |\omega_j(z_j)|<1 \right\}$$ where each $\omega_j$ is univariate.

In the broader Banach space context, for $(X, \|\cdot\|_X)$ a complex Banach sequence space of dimension $n$ (e.g., $\ell_t^n$), the open unit ball is $B_X = \{ z \in X : \|z\|_X < 1 \}$. Holomorphic functions $F:B_X \to Y$ (with $Y$ a complex Banach space) admit Fréchet–Taylor expansions involving symmetric $s$-linear maps $D^sF(0)$ and multi-index power series as above (Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).

2. Multivariate Bohr and Bohr–Rogosinski Inequalities

The classical Bohr inequality for $f(z) = \sum_{k\ge 0} a_k z^k$ holomorphic on $\mathbb{D}$ with $|f(z)|\leq 1$ states

$$\sum_{k\ge 0} |a_k|r^k \leq 1 \quad\text{for}\quad r\leq 1/3$$

with sharpness at $r=1/3$. In several variables, for $f$ holomorphic on $\mathbb{D}^n$ with $|f(z)|\leq 1$, the multivariate Bohr inequality is (Ahamed et al., 10 Jan 2026): $$\sum_{|\alpha|\ge0} |a_\alpha| r^{|\alpha|} \leq 1 \quad\text{for}\quad r \leq R_n := 1/(3n)$$ and $R_n=1/(3n)$ is sharp.

The Bohr–Rogosinski inequality in $\mathbb{D}^n$ is formulated as follows: For $\omega \in \mathcal{B}_{n,m}$ and $N\in\mathbb{N}$,

$$|f(\omega(z))| + \sum_{i=1}^\infty \sum_{|\alpha|=iN} |a_\alpha| r^{|\alpha|} \leq 1 \quad\text{if}\quad n\,r \leq R_{m,n,N}$$

where $R_{m,n,N}$ is the unique positive root of

$$\Psi_{m,n,N}(r) := 2(nr)^N(1 + r^m) - (1 - n r)(1 - r^m) = 0$$

with optimality verified by extremal functions. As $N\to\infty$, $R_{m,n,N}\to 1$ for $n=1$ and $1/n$ for $n\ge 2$ (Ahamed et al., 10 Jan 2026).

In vector-valued and Banach-space settings, analogous Bohr and Bohr–Rogosinski radii are defined, involving sums of norms of Fréchet–Taylor coefficients and their partials, again with sharp values determined by roots of explicit balancing equations incorporating functionally refined terms, parameters for lacunarity, and operator-valued contexts (Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).

3. Derivative and Area-Type Refinements

The multivariable Euler (radial) derivative is given by

$$Df(z) = \sum_{k=1}^n z_k \frac{\partial f}{\partial z_k}(z)$$

which generalizes $z f'(z)$ from the univariate setting. For $f$ bounded by $1$ on $\mathbb{D}^n$,

$|Df(z)| \leq \frac{1 - |f(z)|^2}{1 - r^2} n r$

for $\|z\|_\infty = r$. This estimate sharpens the Bohr inequality further by incorporating local growth via $Df(z)$.

A sharp Bohr plus radial-derivative inequality holds (Ahamed et al., 10 Jan 2026): $$|f(z)| + |Df(z)| + \lambda\sum_{k=2}^\infty \sum_{|\alpha|=k} |a_\alpha| r^{|\alpha|} \leq 1 \quad\text{whenever}\quad n\,r\leq R_{n,\lambda}$$ $R_{n,\lambda}$ is determined exactly as the positive root of a quartic polynomial in $n r$ (distinct forms depending on $\lambda$), with all constants sharp.

Area-based refinements involve the Dirichlet-type sum

$$S_r = \sum_{k\ge1} k\left(\sum_{|\alpha|=k} |a_\alpha|^2\right) r^{2k}$$

and validate inequalities blending Bohr-type and Dirichlet-type terms, extending the classical one-variable “area additive” improvements (Ahamed et al., 10 Jan 2026).

4. Sharpness Mechanisms and Extremal Functions

Establishment of sharpness across these phenomena is achieved by constructing explicit extremal mappings: $$f_a(z) = \frac{a-(z_1+\cdots+z_n)}{1 - a(z_1+\cdots+z_n)},\quad a\in[0,1)$$ For the vector-valued case, extremals reduce to

$$F(z) = \left(\frac{b+z_1}{1+b z_1}, 0, \dots, 0\right),\quad b\to 1^-$$

Testing inequalities at $z = (r,0,\dots,0)$ or on the diagonal $z = (r,\dots, r)$ with $a\to 1$ or $b\to 1$ exposes the limiting behavior at the radius and justifies that no larger $r$ can universally hold (Ahamed et al., 10 Jan 2026, Ahammed et al., 26 Aug 2025). In lacunary or Banach space settings, similar extremals reduce the analysis to the univariate case, retaining the sharp radius.

5. Role of Lacunarity and Functional-Type Corrections

Functions with lacunary expansions

$f(z) = \sum_{s=0}^\infty a_{q s + m} z^{qs + m}$

exhibit improved Bohr radii: the sparsity of nonzero coefficients accelerates convergence of the tail, raising the critical $r$. The Bohr radius becomes the unique solution of

$\frac{r^m}{1 - r^q} = \frac{p}{2}$

or related equations parameterized by the lacunarity pattern and refinement parameters (Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).

Functional-type and norm-type refinements further incorporate the action of supporting functionals in Banach spaces, as well as power corrections and quadratic (area-type) tail terms. These terms do not diminish the sharp radii in finite or infinite dimensions, a robustness property of the phenomenon.

6. Comparison with Classical Univariate Results

In the classical one-variable scenario, the Bohr radius is $1/3$ and the limiting Bohr–Rogosinski radius for the $N$th partial sum is $1$ for large $N$, with $R_{1}=1/3$ and $R_N\uparrow 1$. In several complex variables, the polydisc Bohr radius scales as $1/(3n)$, reflecting the combinatorial growth in the number of homogeneous monomials of fixed degree. Extremal phenomena and proof techniques fundamentally reduce to optimized one-variable behavior on slices, or arguments via Hahn–Banach separation in infinite-dimensional normed settings (Ahamed et al., 10 Jan 2026, Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).

7. Significance and Contemporary Developments

The determination of explicit, sharp Bohr-Rogosinski radii for multivariate, vector-valued, and lacunary holomorphic mappings on domains such as the polydisc or Banach balls resolves longstanding open questions and enables direct transfer of the Bohr phenomenon to multidimensional and operator-theoretic settings. The robustness of these phenomena under addition of functional corrections and their invariance under passage to Banach space geometries highlight the structural depth of the underlying analytic inequalities. Current research also generalizes to directional derivatives, area-type improvements, and explores connections to Dirichlet forms and operator-valued holomorphic function theory (Ahamed et al., 10 Jan 2026, Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).