Area-Based Bohr Inequality Refinements

• Area-Based Bohr Inequality is a refinement of the classical Bohr phenomenon that incorporates two-dimensional Dirichlet area integrals to yield sharper coefficient estimates.
• The methodology unifies analytic, harmonic, and weighted settings by employing majorant series analysis, precise Dirichlet bounds, and extremal Möbius functions.
• It drives further research in operator theory and higher order area functionals, improving classical bounds and enhancing our understanding of mapping behaviors.

The area-based Bohr inequality is a modern refinement of the classical Bohr phenomenon in complex analysis, introducing two-dimensional area measures (typically the Dirichlet area integral) into inequalities for analytic and harmonic mappings on the unit disk. This extension leads to strictly stronger forms of the original coefficient-based results, sharp radii (Bohr radii) for various classes, and unified frameworks for analytic, harmonic, and weighted settings. Area-based Bohr inequalities have been established through an overview of majorant series analysis, sharp bounds on Dirichlet integrals, variational extremal families, and operator-theoretic perspectives.

1. Classical and Refined Bohr Inequalities

The classical Bohr inequality for bounded analytic functions $f(z) = \sum_{n=0}^\infty a_n z^n$ with %%%%1%%%% on $D = \{z\in \mathbb C: |z|<1\}$ states that

$$|a_0| + \sum_{n=1}^\infty |a_n| r^n \leq 1, \quad \text{for } 0 \leq r \leq 1/3,$$

where $1/3$ is sharp (the Bohr radius). Subsequent work has sought either to enlarge the class of functions considered, replace the basis (e.g., with harmonic coefficients), incorporate geometric measures, or strengthen the left side using "area terms" such as the Dirichlet integral

$S_r(f) = \int_{|z|<r} |f'(z)|^2\, dA(z).$

Key improvements take the form

$$|a_0| + \sum_{n=1}^\infty |a_n| r^n + \frac{S_r(f)}{\pi} \leq 1, \quad 0 \leq r \leq 1/3,$$

with extremality realized by the Möbius family $f_a(z) = \frac{a + z}{1 + a z}$ as $a \nearrow 1$ (Ahamed et al., 26 Oct 2025, Ahamed et al., 2024).

2. Area and Quadratic Area Functional

A fundamental aspect of area-based Bohr inequalities is the explicit calculation and estimation of the Dirichlet area integral in terms of the Taylor coefficients: $$S_r(f) = \pi \sum_{n=1}^\infty n |a_n|^2 r^{2n},\qquad \frac{S_r(f)}{\pi} = \sum_{n=1}^\infty n |a_n|^2 r^{2n}.$$ Kayumov and Ponnusamy established the sharp estimate

$$\frac{S_r(f)}{\pi} \leq r^2 \frac{(1 - |a_0|^2)^2}{(1 - |a_0|^2 r^2)^2},\qquad 0<r\leq 1/\sqrt{2},$$

enabling optimal inclusion of $S_r(f)$ in refined Bohr inequalities (Ahamed et al., 26 Oct 2025). This area term can be further iterated to quadratic (and higher) powers, yielding inequalities such as

$$B_0(f, r) + \frac{16}{9} \frac{S_r}{\pi} + \lambda_1\left(\frac{S_r}{\pi}\right)^2 \leq 1, \quad 0 \leq r \leq 1/3$$

with sharp $\lambda_1$ and more stringent radii (Aahmed et al., 4 Dec 2025). Alternative normalizations such as $S_r/(π−S_r)$ yield analogous results with adjusted multipliers and radii (Aahmed et al., 4 Dec 2025, Ahamed et al., 2024).

3. Unification with Harmonic and Weighted Settings

Modern developments extend area-based Bohr inequalities to harmonic mappings $f = h + \overline{g}$, where both analytic and anti-analytic components are present, using frameworks such as

$$A_f(r) = |z|\, \varphi_0(r) + \sum_{n=2}^\infty (|a_n| + |b_n|) \varphi_n(r),$$

where $\{\varphi_n(r)\}$ is a sequence of nonnegative differentiable, increasing functions on $[0,1)$. The resulting sharp Bohr-type inequalities involve estimating

$$r\,\varphi_0(r) + 2M\,\sum_{n=2}^\infty \frac{\varphi_n(r)}{n(n-1)} \leq L_M(1)$$

for the class $\mathcal{P}^0_{\mathcal{H}(M)}$, or similar expressions for $\mathcal{W}^0_{\mathcal{H}(\alpha)}$ (Ahamed et al., 26 Oct 2025). Weightings such as $\varphi_n(r) = n r^n$ or $n^2 r^n$ allow the construction of weighted area-based Bohr radii, further broadening the spectrum of applicable inequalities.

4. Multi-Parameter and Extremal Family Methodology

A salient feature of the area-based approach is the explicit identification of extremal functions and the sharpness analysis. In the setting of bounded analytic functions $f(z) = \sum_{n=0}^\infty a_n z^n\in H^\infty$, and for any fixed $m\in \mathbb{N}$ with auxiliary Schwarz function $\omega(z)$ (with $\omega(0) = \dots = \omega^{(m-1)}(0) = 0$), a suite of generalizations is proved: $$A_f(z) = |f(\omega(z))| + B_1(f, r) + \frac{1+a r}{(1+a)(1-r)}|f_0|_r \leq 1 \quad \text{for } |z| = r \leq \alpha_m,$$ where $\alpha_m$ is the unique root of $(1-r)(1-r^m) - 2r(1 + r^m) = 0$. As $m\to\infty$, the radii converge to the classical values (e.g., $1/3$), interpolating between the refined and the original Bohr setting. The principal extremal family remains $f_a(z)$ and $\omega(z)=z^m$ (Huang et al., 2020).

In the analytic, harmonic, or weighted area-based context, sharpness is always demonstrated by matching the inequality at the endpoint radius using explicit extremal mappings, most prominently the Möbius functions. This ensures that all constants, multipliers, and radii in the inequalities are optimal and non-improvable.

5. Operator-Theoretic and Functional Perspectives

Recent work introduces operator-theoretic frameworks, such as defining the area-based Bohr operator

$B_r[f] = \sum_{n=0}^\infty |a_n| r^n,$

and analyzing its behavior under function-theoretic operations, pre-compositions, and subordinations. Results for sections $s_k(f)$ and pre-compositions with Schwarz maps establish sharp operator norms and identify invariance or contraction properties up to $r=1/3$ (Huang et al., 2021).

A typical area-based Bohr inequality formulated in this spirit includes terms such as

$$\sum_{n=0}^\infty |a_n| r^n + \frac{1}{1+|a_0|} \sum_{n=1}^\infty |a_n| r^n + \frac{8}{9\pi} S_r(f) + \lambda \frac{r^4}{(1 - |a_0| r)^4} \leq 1$$

with precisely determined constants from algebraic equations in $a$ and $r$. These approaches unify classical, area-based, and operator perspectives.

6. Proof Techniques and Key Ingredients

The derivation of area-based Bohr inequalities relies on the interplay among:

• Schwarz–Pick lemma for analytic self-maps, giving distortion and derivative estimates, e.g.,

$|f'(z)| \leq \frac{1 - |f(z)|^2}{1 - |z|^2}$

• Sharp coefficient bounds, $|a_n| \leq 1 - |a_0|^2$ for $n \geq 1$
• Coefficient-majorant and power-series Bohr-sum lemmas, e.g., $\sum_{n \geq 1} |a_n| r^n \leq (1 - a^2)/(1 - a r)$
• Two-dimensional area estimates in terms of Taylor coefficients and their normalization
• Reduction to elementary univariate polynomial or rational inequalities in $r$, sometimes parametrized by $|a_0|$, with unique roots dictating the sharp radii
• Verification of extremality by evaluation at the endpoint for a one-parameter Möbius family

Proofs typically proceed by assembling upper bounds for sum, area, and auxiliary terms into a single real function of $r$, verifying monotonicity, endpoint conditions, and optimality (Huang et al., 2020, Ahamed et al., 2024, Huang et al., 2021).

7. Extensions, Corollaries, and Open Problems

Area-based Bohr inequalities admit numerous corollaries:

• Reverting to the classical Bohr theorem by nullifying area terms, retrieving $r=1/3$
• Weighted, harmonic, and generalized basis extensions yield diverse radii and multipliers
• Rational and quadratic area terms such as $S_r/(π-S_r)$ or $(S_r/\pi)^2$ offer sharper control in more restrictive domains (Aahmed et al., 4 Dec 2025)

Open questions include further extension to larger analytic or harmonic classes; the systematic inclusion of higher order area functionals (cubic, quartic, etc.); and the determination of sharp constants for multidimensional, operator-valued, or quasi-regular cases (Aahmed et al., 4 Dec 2025). The area-based framework thus continues to be an active area of research, with impact in geometric function theory, operator theory, and associated combinatorial analysis.