Bohr Phenomena in Analytic Function Theory

The Bohr phenomenon refers to a striking interplay between the Taylor coefficients of an analytic function and its uniform bound on the unit disc. Originally established in the early 20th century, it asserts that for any function f(z)=∑aₙzⁿ analytic in |z|<1 and bounded by unity, the sum ∑|aₙ|rⁿ remains below one for all radii r up to a critical value known as the Bohr radius. Modern investigations have vastly extended this classical result. In several complex variables one seeks radii independent of the ambient dimension or governed by individual degrees, while in harmonic and log-harmonic settings the decomposition into analytic and co-analytic parts demands refined coefficient inequalities. Quantum‐calculus variants introduce q-starlike and q-convex classes, and noncommutative analogues explore operator‐valued series and quantum Boolean cubes. Further generalisations address alternative metrics, such as the hyperbolic distance, and domains beyond the unit disc. Across these contexts the focus remains on determining sharp radii, elucidating asymptotic behaviour in high dimensions and uncovering links to geometric function theory, operator norms and applications in signal processing and quantum information. The breadth of recent work highlights the Bohr phenomenon as a unifying theme in analytic inequalities, revealing deep structural properties of holomorphic and harmonic mappings under coefficient constraints.

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Recent studies have refined Bohr‐type radii for specialised classes of harmonic mappings. In particular, the optimal radius for close-to-starlike harmonic mappings has been determined, and the classical Bohr–Rogosinski phenomenon has been extended to these classes, revealing exact bounds that depend on prescribed stability parameters. Complementing this, new sharp inequalities for bounded analytic functions subordinate to Schwarz functions have been established, introducing one-parameter and convex-combination forms of the Bohr inequality. These results generalise earlier bounds and demonstrate the role of subordination in controlling coefficient sums. In the noncommutative realm, advances in Bohnenblust–Hille inequalities on quantum Boolean cubes have uncovered dimension-free Bohr radii with exponential growth rates in the degree. Such operator‐theoretic developments not only bridge classical analytic theory and quantum learning problems but also illustrate the universality of Bohr’s radius phenomenon across both commutative and noncommutative function spaces.

Bohr Phenomena in Analytic Function Theory publication trend

The graph below shows the total number of articles in bohr phenomena in analytic function theory across all publications each year (not limited to Nature Index journals).

Technical terms

Bohr radius: The largest radius R such that the coefficient sum ∑|aₙ|rⁿ of an analytic function remains bounded by its sup-norm for all r≤R.

Harmonic mapping: A complex-valued function on the unit disc satisfying Laplace’s equation, expressible as the sum of an analytic part and a conjugate analytic part.

Schwarz function: An analytic self-map of the unit disc that fixes the origin, often used to define subordination relations between analytic functions.

Subordination: A relation f≺g indicating that f(z)=g(ω(z)) for some Schwarz function ω, ensuring that f inherits certain bounds from g.

Starlike function: An analytic univalent function on the unit disc whose image is a star-shaped domain with respect to the origin, characterised by Re(zf′(z)/f(z))>0.

Bohr–Rogosinski phenomenon: A refinement combining the classical Bohr inequality with pointwise estimates, yielding joint control of partial sums and function values within a radius.

References

1. Bohr Radius Problems for Some Classes of Analytic Functions Using Quantum Calculus Approach. Mathematics (2020).
2. Bohr-type inequalities for bounded analytic functions of Schwarz functions. AIMS Mathematics (2021).
3. Bohr radius and its asymptotic value for holomorphic functions in higher dimensions. Comptes Rendus Mathématique (2021).
4. Noncommutative Bohnenblust–Hille inequalities. Mathematische Annalen (2023).
5. On Bohr's inequality for special subclasses of stable starlike harmonic mappings. Open Mathematics (2023).
6. Bohr’s inequality for analytic functions with hyperbolic range. Mathematics Open (2025).