Bohr Radius in Complex Analysis

• Bohr radius is the maximum radius within which the absolute sum of power series coefficients remains bounded by the function's supremum, classically equal to 1/3 for analytic functions on the unit disk.
• It has been extended to polynomial, harmonic, and operator-valued mappings, with optimal radii determined through methods like Toeplitz determinants and spectral analysis.
• Applications span several complex variables and Banach space theory, where refined asymptotic estimates and mixed radii reveal the nuanced impact of domain and codomain changes on functional bounds.

The Bohr radius is the extremal radius governing when the coefficient majorant of a holomorphic expansion remains controlled by the ambient boundedness of the function. In its classical form, if $f(z)=\sum_{n=0}^{\infty} a_n z^n$ is analytic on the unit disk $\mathbb D$ and $\|f\|_{\infty}\le 1$, the Bohr inequality asks for the largest $r\in(0,1)$ such that $\sum_{n=0}^{\infty}|a_n|r^n\le 1$. That sharp radius is $1/3$. Subsequent work has turned this scalar one-variable constant into a broad family of extremal radii for polynomials, harmonic mappings, subordinate classes, several-complex-variable domains, operator-valued functions, and basis-dependent expansions, with techniques ranging from Schwarz–Pick estimates to Toeplitz determinants, unconditional basis constants, and local Banach space theory (Muhanna et al., 2016, Chu, 2014, Halder, 22 Dec 2025).

1. Classical theorem and extremal structure

For $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$, Bohr’s theorem states that if

$$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$

then

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$

and $1/3$ is optimal. A standard proof uses the sharp coefficient bound

$\mathbb D$0

deduced from Schwarz–Pick, followed by the estimate

$\mathbb D$1

whose optimization in $\mathbb D$2 yields the threshold $\mathbb D$3. Sharpness is exhibited by Möbius or Blaschke extremals, such as $\mathbb D$4, with $\mathbb D$5 (Muhanna et al., 2016).

Historically, Harald Bohr introduced the inequality in 1914 in connection with the absolute convergence of Dirichlet series $\mathbb D$6. The first version had radius $\mathbb D$7; the sharp constant $\mathbb D$8 was subsequently obtained independently by M. Riesz, I. Schur, and N. Wiener. The result also became part of the modern structure of almost periodic functions and geometric function theory (Muhanna et al., 2016).

A common misconception is that the value $\mathbb D$9 is intrinsic to the phrase “Bohr radius” in every setting. In fact, $\|f\|_{\infty}\le 1$0 is only the classical scalar $\|f\|_{\infty}\le 1$1 constant. Once the coefficient geometry, codomain, domain, or function class is changed, the sharp radius may increase, decrease, or even decay asymptotically with dimension.

2. Polynomial Bohr radius and the asymptotic regime

For the polynomial subspace

$\|f\|_{\infty}\le 1$2

the polynomial Bohr radius $\|f\|_{\infty}\le 1$3 is the largest $\|f\|_{\infty}\le 1$4 such that

$\|f\|_{\infty}\le 1$5

for every $\|f\|_{\infty}\le 1$6. One has $\|f\|_{\infty}\le 1$7, but $\|f\|_{\infty}\le 1$8 is strictly larger than the classical radius and approaches $\|f\|_{\infty}\le 1$9 as $r\in(0,1)$0 (Chu, 2014).

A decisive characterization is due to Fournier: $r\in(0,1)$1 is exactly the smallest $r\in(0,1)$2 for which

$r\in(0,1)$3

where $r\in(0,1)$4 is an explicit $r\in(0,1)$5 symmetric Toeplitz matrix with diagonal entries $r\in(0,1)$6 and alternating signed powers of $r\in(0,1)$7 off the diagonal. This converts the Bohr-radius problem into a spectral problem for Toeplitz determinants (Chu, 2014).

The asymptotic formula proved for $r\in(0,1)$8 is

$r\in(0,1)$9

This confirms Fournier’s conjecture and sharpens earlier coarse bounds. The proof proceeds through a determinant recurrence

$\sum_{n=0}^{\infty}|a_n|r^n\le 1$0

a trigonometric reparametrization using the symbol

$\sum_{n=0}^{\infty}|a_n|r^n\le 1$1

and an analysis of the largest zero $\sum_{n=0}^{\infty}|a_n|r^n\le 1$2 of an associated sine-quotient polynomial $\sum_{n=0}^{\infty}|a_n|r^n\le 1$3. Writing $\sum_{n=0}^{\infty}|a_n|r^n\le 1$4, one obtains $\sum_{n=0}^{\infty}|a_n|r^n\le 1$5, and substitution into $\sum_{n=0}^{\infty}|a_n|r^n\le 1$6 yields the stated expansion (Chu, 2014).

This asymptotic result is significant because it shows that the polynomial constraint changes the Bohr radius at second order rather than first order. The correction term is explicit and universal, and the proof is an example of how a coefficient-majorant problem can be resolved by determinant asymptotics.

3. Harmonic mappings and related radii

The Bohr phenomenon extends from analytic maps to harmonic mappings

$\sum_{n=0}^{\infty}|a_n|r^n\le 1$7

typically with $\sum_{n=0}^{\infty}|a_n|r^n\le 1$8. For bounded harmonic mappings $\sum_{n=0}^{\infty}|a_n|r^n\le 1$9 in $1/3$0, a sharp coefficient estimate is

$1/3$1

with equality attained by rotations of a harmonic Koebe-type map. If $1/3$2, then the majorant series satisfies

$1/3$3

so the sharp Bohr radius is

$1/3$4

In particular, when $1/3$5,

$1/3$6

The same framework extends to harmonic Poisson integrals $1/3$7 with $1/3$8: if $1/3$9, then the sharp radius is

$$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$0

where

$$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$1

Under the standard normalization $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$2, $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$3, $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$4, the same coefficient control yields an explicit radius of univalence

$$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$5

and a radius of the inscribed schlicht disk

$$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$6

These constants are presented as best possible (Ahamed et al., 12 Apr 2026).

Other harmonic classes exhibit different sharp thresholds. For sense-preserving harmonic maps with bounded analytic part $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$7, the Bohr radius is $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$8, sharp; if both $$H^\infty=\{f:\mathbb D\to\mathbb C\text{ holomorphic},\ \|f\|_\infty<\infty\}$$9 and $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$0 are bounded, the sharp radius becomes $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$1; if $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$2, the admissible radius is the unique root $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$3 of

$$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$4

For analytic Bloch functions and harmonic Bloch functions, a Bohr-type radius $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$5 is determined by the equation

$$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$6

(Kayumov et al., 2017).

For close-to-convex harmonic mappings in the class $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$7, the theory further branches into Bohr–Rogosinski, improved, and refined radii, each defined by explicit root equations and each increasing with the parameter $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$8. The extremal map

$$f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \|f\|_\infty\le 1,$$9

governs the sharpness statements (Ahamed et al., 2020).

4. Subordination, geometric function classes, and special target domains

A large part of the modern literature reformulates the Bohr phenomenon as a distance-to-boundary estimate: $$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$0 For Janowski-starlike functions $$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$1, defined by

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$2

the sharp Bohr radius is characterized as the unique root of an explicit coefficient-growth equation involving the sums

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$3

A parallel formula holds for second-order differential subordinations

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$4

with the denominator $$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$5 appearing in the coefficient bound. For typically real functions, the sharp radius is

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$6

obtained from the equation

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$7

(Anand et al., 2020).

For Ma–Minda starlike and convex classes, the Bohr radii are determined by the Koebe-type generators

$$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$8

If $$\sum_{n=0}^{\infty}|a_n|\,r^n\le 1\qquad \text{for }0\le r\le \frac13,$$9, then the radius is $1/3$0, where $1/3$1 is the unique positive root of

$1/3$2

If $1/3$3, the analogous radius is $1/3$4, with

$1/3$5

For the class $1/3$6 of functions starlike with respect to a boundary point, the radius is closed-form: $1/3$7 which reduces to $1/3$8 at $1/3$9 and tends to $\mathbb D$00 as $\mathbb D$01 (Allu et al., 2020).

Further refinements introduce additional Schwarz-function or area terms. In the degenerate Janowski case $\mathbb D$02, the sharp improved Bohr radius can be encoded by an equation involving the Bessel function $\mathbb D$03, for example

$\mathbb D$04

showing that even within a fixed subordinate family the sharp threshold can depend sensitively on the chosen refinement (Ahamed et al., 2024).

The survey literature also records Bohr radii for image domains beyond the disk. For subordination to a general univalent image $\mathbb D$05, the subordination radius $\mathbb D$06 appears. For concave wedge-domains

$\mathbb D$07

the sharp radius is

$\mathbb D$08

For the punctured disk, the exterior of the closed unit disk, and half-plane-type targets, analogous inequalities are formulated using spherical chordal or hyperbolic metrics, and $\mathbb D$09 frequently reappears in those metric versions (Muhanna et al., 2016).

5. Several complex variables, mixed radii, and Banach-space asymptotics

In several complex variables, the Bohr radius is defined on complete Reinhardt domains. If

$\mathbb D$10

is holomorphic on a complete Reinhardt domain $\mathbb D$11 with $\mathbb D$12, the $\mathbb D$13-dimensional Bohr radius $\mathbb D$14 is the largest $\mathbb D$15 such that

$\mathbb D$16

on the homothetic copy $\mathbb D$17. For the unit polydisk $\mathbb D$18, the classical estimates are

$\mathbb D$19

so $\mathbb D$20 at the slow rate $\mathbb D$21 (Muhanna et al., 2016).

The mixed $\mathbb D$22-Bohr radius

$\mathbb D$23

admits a full asymptotic classification. As $\mathbb D$24, if $\mathbb D$25 and $\mathbb D$26, then

$\mathbb D$27

If $\mathbb D$28, then

$\mathbb D$29

If $\mathbb D$30, then

$\mathbb D$31

and for $\mathbb D$32 one likewise has $\mathbb D$33 (Galicer et al., 2017).

Vector-valued refinements sharpen these asymptotics. For bounded holomorphic maps from $\mathbb D$34 into a finite-dimensional Banach space $\mathbb D$35, improved lower estimates remove the extra $\mathbb D$36 factor present in earlier bounds, giving

$\mathbb D$37

for suitable $\mathbb D$38 (Allu et al., 30 Jun 2025).

The arithmetic Bohr radius extends the theory to operator-valued pluriharmonic functions on complete Reinhardt domains. For a bounded linear map $\mathbb D$39 and $\mathbb D$40, the powered and arithmetic Bohr radii $\mathbb D$41 and $\mathbb D$42 measure coefficient control either through supremal dilations or through coordinatewise radii. When $\mathbb D$43 is the unit ball of a finite-dimensional Banach space $\mathbb D$44 with $\mathbb D$45-unconditional basis, one has

$\mathbb D$46

and for classical $\mathbb D$47-balls the asymptotic scale is again of order $\mathbb D$48. The same framework extends to mixed Minkowski, Lorentz, and Orlicz sequence spaces (Halder, 22 Dec 2025).

6. Operator-theoretic, basis-dependent, and nonclassical formulations

A further generalization replaces the function class by a pair of operators. If $\mathbb D$49 and $\mathbb D$50 act on analytic power-series spaces, the Bohr radius $\mathbb D$51 is the largest $\mathbb D$52 such that $\mathbb D$53 for $\mathbb D$54, where $\mathbb D$55 is the coefficient majorant. For Hadamard convolution operators this produces a general Bohr–Bombieri function, from which exact radii for differentiation and integration follow. In particular,

$\mathbb D$56

For the Volterra integration operator, an explicit formula involving the Lambert $\mathbb D$57-function is obtained when $\mathbb D$58, and one also has the sharp bounds

$\mathbb D$59

(Khasyanov, 2023).

For weighted Bloch spaces $\mathbb D$60, every nonnegative radial weight $\mathbb D$61 satisfies the universal lower bound

$\mathbb D$62

Sharpness at $\mathbb D$63 is characterized by an explicit inequality involving a distinguished point $\mathbb D$64, and concrete extremal weights include the constant weight, piecewise power-type weights, and Möbius-type weights (Khasyanov, 2023).

The basis-dependent nature of the Bohr phenomenon is especially clear for Faber expansions on condensers. For the elliptic condenser associated with $\mathbb D$65, the conformal map satisfies

$\mathbb D$66

and the Faber polynomials take the form

$\mathbb D$67

The exact Bohr radius is no longer $\mathbb D$68: in the full complex-coefficient case,

$\mathbb D$69

while for real coefficients,

$\mathbb D$70

These values are determined by explicit infinite-series equations in $\mathbb D$71 (Lassère et al., 2011).

Related Banach-valued theories on simply connected domains introduce weighted radii $\mathbb D$72 and recover classical-type constants in shifted disks. For operator-valued functions on $\mathbb D$73, the classical weight $\mathbb D$74 yields the sharp radius

$\mathbb D$75

and analogous root equations govern Cesàro and Bernardi transforms (Allu et al., 2021).

Taken together, these developments show that the Bohr radius is best understood not as a single universal constant, but as an extremal threshold attached to a coefficient geometry. Its value depends on the domain of holomorphy, the target geometry, the function class, the basis, the operator acting on coefficients, and, in high dimension, the ambient Banach-space structure itself.