Bohr-Type Inequalities in Complex Analysis

• Bohr-type inequalities are refined analytic inequalities that control sums of coefficients, derivatives, or area terms within a subradius of the unit disk.
• They extend the classical Bohr radius by incorporating quadratic, weighted, and operator-theoretic corrections, with sharp radii determined by explicit equations.
• Advances in this area impact analytic, harmonic, and multivariable frameworks, providing precise bounds in diverse settings such as fractional calculus and Banach spaces.

Bohr-type inequalities are inequalities that control a majorant built from coefficients, function values, derivatives, Jacobians, or area terms by a fixed bound, a boundary-distance quantity, or an operator-specific majorant on a subdisk or analogous domain. Their prototype is the classical theorem: if

$f(z)=\sum_{n=0}^{\infty} a_n z^n$

is analytic in $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $|f(z)|\le 1$ in $\mathbb D$, then

$$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$

and $1/3$ is sharp. In current usage, the term encompasses a much broader family: refined analytic inequalities with quadratic and area corrections, Bohr–Rogosinski variants, harmonic and quasiconformal analogues, weighted and operator-theoretic forms, multivariable and Banach-space extensions, slice regular and fractional versions, and class-specific results for close-to-convex, convex, univalent, or lacunary families (Ponnusamy et al., 2019, Ahamed et al., 2021, Fujii et al., 2011).

1. Classical formulation and foundational viewpoints

In one complex variable, the basic setting is the bounded analytic class

$$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$

For $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$, the classical Bohr inequality is often written in the equivalent form

$$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$

with sharp radius $1/3$. Two standard refinements already alter the radius without changing the class: replacing $\mathbb D=\{z\in\mathbb C:|z|<1\}$0 by $\mathbb D=\{z\in\mathbb C:|z|<1\}$1 gives radius $\mathbb D=\{z\in\mathbb C:|z|<1\}$2, while imposing $\mathbb D=\{z\in\mathbb C:|z|<1\}$3 gives sharp radius $\mathbb D=\{z\in\mathbb C:|z|<1\}$4 (Ponnusamy et al., 2019).

A separate but historically related usage appears in operator theory. There the scalar inequality

$\mathbb D=\{z\in\mathbb C:|z|<1\}$5

is treated as the classical Bohr inequality, and it admits operator, matrix-order, Jensen-type, and eigenvalue generalizations. This branch includes inequalities such as

$\mathbb D=\{z\in\mathbb C:|z|<1\}$6

which the literature interprets as an operator Jensen inequality for $\mathbb D=\{z\in\mathbb C:|z|<1\}$7 (Fujii et al., 2011).

These two viewpoints share a common structure: a nontrivial majorant is controlled on a smaller region than the original domain of boundedness, and the maximal admissible radius is part of the theorem. In function theory that radius is the Bohr radius; in operator-theoretic forms it is replaced by matrix-order or convexity constraints. This suggests that “Bohr-type” is best understood as a radius-sensitive majorization principle rather than a single fixed statement.

2. Refined inequalities for bounded analytic functions

A major development replaces the bare coefficient sum by expressions containing positive correction terms. One direction uses quadratic coefficient functionals. For $\mathbb D=\{z\in\mathbb C:|z|<1\}$8,

$\mathbb D=\{z\in\mathbb C:|z|<1\}$9

where $|f(z)|\le 1$0. Further sharp inequalities of the form

$|f(z)|\le 1$1

hold for $|f(z)|\le 1$2, and related variants hold for $|f(z)|\le 1$3, $|f(z)|\le 1$4, or $|f(z)|\le 1$5, depending on the normalization and the quadratic term used (Ponnusamy et al., 2019).

Another direction adds area-type corrections. For bounded analytic functions, one has sharp inequalities such as

$|f(z)|\le 1$6

where $|f(z)|\le 1$7 is the Euclidean area of $|f(z)|\le 1$8. The same paper also recalls the sharp earlier estimate

$|f(z)|\le 1$9

showing that replacing $\mathbb D$0 by $\mathbb D$1 restores the classical radius $\mathbb D$2 (Ismagilov et al., 2020).

Area refinements became more delicate in later work. For instance, a stronger nonlinear correction replaces $\mathbb D$3 by

$\mathbb D$4

leading to sharp inequalities such as

$\mathbb D$5

and

$\mathbb D$6

The same line of work also proves sharper parameterized versions involving $\mathbb D$7, $\mathbb D$8, and explicit sharp constants $\mathbb D$9 determined by algebraic equations (Ahamed et al., 2023).

Vanishing order at the origin produces another class of refinements. For

$$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$0

sharp inequalities include

$$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$1

for $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$2, where $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$3 is the unique root of

$$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$4

A second sharp variant uses

$$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$5

for $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$6, where $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$7 is the unique solution of

$$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$8

These results show that prescribed multiple zeros improve the admissible radius in a structured way (Ponnusamy et al., 2020).

3. Weighted, parameterized, and operator-transform formulations

A modern branch of the subject replaces the monomial weights $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (r\le 1/3),$$9 by more general operator- or weight-dependent terms. For the Cesàro operator,

$1/3$0

the associated Bohr sum is

$1/3$1

If $1/3$2, then

$1/3$3

where $1/3$4 is the positive root of

$1/3$5

This is sharp. The same paper gives sharp radii for the $1/3$6-Cesàro, Bernardi, Libera, and Alexander operators, with the Libera/Alexander radius

$1/3$7

defined by

$1/3$8

(Kumar et al., 2020).

The Cesàro theory has also been refined by inserting derivative data. If $1/3$9, then

$$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$0

for $$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$1, where $$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$2 is the unique positive root of the explicit transcendental equation displayed in the paper. This is sharp, and the extremal function is the Möbius map

$$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$3

with $$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$4 (Allu et al., 2024).

Weighted and parameterized frameworks generalize the majorant itself. One approach introduces two parameters $$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$5 and proves inequalities such as

$$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$6

for $$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$7, where $$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$8 is the positive root of

$$\mathcal B=\{f \text{ analytic in } \mathbb D:\ |f(z)|\le 1 \text{ for all } z\in\mathbb D\}.$$9

Another introduces parameters $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$0 and proves

$f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$1

with

$f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$2

These families recover classical and one-parameter Bohr inequalities as special cases (Zhou et al., 5 Feb 2025, Hou et al., 5 Feb 2025).

A still more flexible generalization replaces $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$3 by a weight sequence $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$4, where

$f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$5

In this setting the papers study inequalities for $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$6, where $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$7 is a Schwarz function with $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$8-fold zero, and derive sharp radii from conditions such as $f(z)=\sum_{n=0}^\infty a_n z^n\in\mathcal B$9, $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$0, and $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$1. This single-parameter framework with $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$2 subsumes several earlier results when $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$3 (Chen et al., 2023).

4. Harmonic, quasiconformal, and multiple-zero settings

For harmonic mappings $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$4 in $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$5, Bohr-type theory must account for the coupled analytic and co-analytic coefficients. A central class is

$$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$6

with normalized series

$$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$7

For $$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$8, functions in this class are fully starlike; the sharper range

$$\sum_{n=1}^\infty |a_n|\,r^n \le 1-|a_0| \qquad (r\le 1/3),$$9

ensures positivity of

$1/3$0

The coefficient theory is explicit: $1/3$1 with extremal function

$1/3$2

The associated boundary-distance estimate is

$1/3$3

and equality is attained by $1/3$4 (Ahamed et al., 2021).

On this class one obtains several sharp Bohr-type inequalities. The Bohr–Rogosinski form is

$1/3$5

for $1/3$6, where $1/3$7 is the unique root of

$1/3$8

There are also sharp refinements involving the area

$1/3$9

the Jacobian

$\mathbb D=\{z\in\mathbb C:|z|<1\}$00

and a square-majorant form involving

$\mathbb D=\{z\in\mathbb C:|z|<1\}$01

Each radius is defined as the unique root of an explicit equation and is best possible (Ahamed et al., 2021).

A more general two-parameter harmonic class is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$02

defined by the second-order differential inequality

$\mathbb D=\{z\in\mathbb C:|z|<1\}$03

Here the coefficient bounds are

$\mathbb D=\{z\in\mathbb C:|z|<1\}$04

and one sharp improved Bohr inequality is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$05

For $\mathbb D=\{z\in\mathbb C:|z|<1\}$06, the paper reports

$\mathbb D=\{z\in\mathbb C:|z|<1\}$07

and area-refined radii

$\mathbb D=\{z\in\mathbb C:|z|<1\}$08

all best possible (Wang et al., 25 May 2026).

Multiple zeros at the origin generate another harmonic branch. For $\mathbb D=\{z\in\mathbb C:|z|<1\}$09-symmetric harmonic mappings with

$\mathbb D=\{z\in\mathbb C:|z|<1\}$10

the literature proves sharp Bohr-type inequalities under either boundedness of both parts or the weaker domination

$\mathbb D=\{z\in\mathbb C:|z|<1\}$11

Representative sharp radii are defined by equations such as

$\mathbb D=\{z\in\mathbb C:|z|<1\}$12

$\mathbb D=\{z\in\mathbb C:|z|<1\}$13

and

$\mathbb D=\{z\in\mathbb C:|z|<1\}$14

with extremals of the form

$\mathbb D=\{z\in\mathbb C:|z|<1\}$15

or $\mathbb D=\{z\in\mathbb C:|z|<1\}$16 (Huang et al., 2021).

For $\mathbb D=\{z\in\mathbb C:|z|<1\}$17-quasiconformal harmonic mappings, recent work introduces multiple Schwarz functions $\mathbb D=\{z\in\mathbb C:|z|<1\}$18 to majorize different terms independently. One sharp result states that, for suitable $\mathbb D=\{z\in\mathbb C:|z|<1\}$19,

$\mathbb D=\{z\in\mathbb C:|z|<1\}$20

for $\mathbb D=\{z\in\mathbb C:|z|<1\}$21, where $\mathbb D=\{z\in\mathbb C:|z|<1\}$22 and $\mathbb D=\{z\in\mathbb C:|z|<1\}$23 is the smallest positive root of

$\mathbb D=\{z\in\mathbb C:|z|<1\}$24

The same framework yields improved inequalities involving $\mathbb D=\{z\in\mathbb C:|z|<1\}$25, $\mathbb D=\{z\in\mathbb C:|z|<1\}$26, and square/area-type corrections (Biswas et al., 1 Oct 2025).

5. Multivariable, Banach-space, slice regular, fractional, and geometric extensions

In several complex variables, refined Bohr theory on the polydisk $\mathbb D=\{z\in\mathbb C:|z|<1\}$27 uses the scaling parameter $\mathbb D=\{z\in\mathbb C:|z|<1\}$28, where $\mathbb D=\{z\in\mathbb C:|z|<1\}$29. If

$\mathbb D=\{z\in\mathbb C:|z|<1\}$30

is holomorphic in the unit polydisk and $\mathbb D=\{z\in\mathbb C:|z|<1\}$31 on $\mathbb D=\{z\in\mathbb C:|z|<1\}$32, then there are sharp multidimensional analogues of refined Bohr–Rogosinski inequalities, $\mathbb D=\{z\in\mathbb C:|z|<1\}$33-Bohr inequalities, and derivative-based results. The radial derivative is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$34

and one sharp multidimensional refinement is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$35

for

$\mathbb D=\{z\in\mathbb C:|z|<1\}$36

and this constant is best possible. Higher-order mixed partials satisfy analogous sharp inequalities with radii determined by

$\mathbb D=\{z\in\mathbb C:|z|<1\}$37

or

$\mathbb D=\{z\in\mathbb C:|z|<1\}$38

(Ahamed et al., 11 Dec 2025).

For holomorphic functions on the unit ball $\mathbb D=\{z\in\mathbb C:|z|<1\}$39 of a finite-dimensional Banach sequence space, the theory extends to lacunary and alternating series. If

$\mathbb D=\{z\in\mathbb C:|z|<1\}$40

then a sharp lacunary Bohr inequality holds for $\mathbb D=\{z\in\mathbb C:|z|<1\}$41, where $\mathbb D=\{z\in\mathbb C:|z|<1\}$42 is the unique root of

$\mathbb D=\{z\in\mathbb C:|z|<1\}$43

In the special case $\mathbb D=\{z\in\mathbb C:|z|<1\}$44,

$\mathbb D=\{z\in\mathbb C:|z|<1\}$45

The same paper proves vector-valued and alternating analogues with sharp radii given by

$\mathbb D=\{z\in\mathbb C:|z|<1\}$46

(Kumar et al., 2024).

Slice regular function theory over the octonions provides a non-associative analogue. For

$\mathbb D=\{z\in\mathbb C:|z|<1\}$47

defined on the unit ball $\mathbb D=\{z\in\mathbb C:|z|<1\}$48, the generalized Bohr inequality reads

$\mathbb D=\{z\in\mathbb C:|z|<1\}$49

for every $\mathbb D=\{z\in\mathbb C:|z|<1\}$50, and $\mathbb D=\{z\in\mathbb C:|z|<1\}$51 is best possible. Refined versions add terms such as $\mathbb D=\{z\in\mathbb C:|z|<1\}$52, square-sums of coefficients, or

$\mathbb D=\{z\in\mathbb C:|z|<1\}$53

In the half-space case $\mathbb D=\{z\in\mathbb C:|z|<1\}$54, the paper proves a sharp radius

$\mathbb D=\{z\in\mathbb C:|z|<1\}$55

defined by

$\mathbb D=\{z\in\mathbb C:|z|<1\}$56

(Ahammed et al., 26 Jun 2025).

Fractional calculus yields another branch. For the Riemann–Liouville derivative

$\mathbb D=\{z\in\mathbb C:|z|<1\}$57

the fractional Bohr sum is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$58

For $\mathbb D=\{z\in\mathbb C:|z|<1\}$59, one obtains a sharp radius $\mathbb D=\{z\in\mathbb C:|z|<1\}$60 determined by

$\mathbb D=\{z\in\mathbb C:|z|<1\}$61

The numerical values reported in the paper are

$\mathbb D=\{z\in\mathbb C:|z|<1\}$62

For the $\mathbb D=\{z\in\mathbb C:|z|<1\}$63-weighted variant the sharp fractional radius $\mathbb D=\{z\in\mathbb C:|z|<1\}$64 satisfies

$\mathbb D=\{z\in\mathbb C:|z|<1\}$65

The paper also gives univalent, convex, and Bloch-function versions, all with $\mathbb D=\{z\in\mathbb C:|z|<1\}$66-dependent sharp radii (Mogbademu et al., 25 Sep 2025).

A geometric subclass approach appears for close-to-convex functions. For the Silverman–Telage classes $\mathbb D=\{z\in\mathbb C:|z|<1\}$67, the literature proves sharp inequalities involving $\mathbb D=\{z\in\mathbb C:|z|<1\}$68, $\mathbb D=\{z\in\mathbb C:|z|<1\}$69, coefficient tails, and $\mathbb D=\{z\in\mathbb C:|z|<1\}$70-power sums. Representative examples are

$\mathbb D=\{z\in\mathbb C:|z|<1\}$71

for $\mathbb D=\{z\in\mathbb C:|z|<1\}$72, where

$\mathbb D=\{z\in\mathbb C:|z|<1\}$73

is the unique solution of an explicit logarithmic equation in $\mathbb D=\{z\in\mathbb C:|z|<1\}$74, and

$\mathbb D=\{z\in\mathbb C:|z|<1\}$75

defined by

$\mathbb D=\{z\in\mathbb C:|z|<1\}$76

in $\mathbb D=\{z\in\mathbb C:|z|<1\}$77 (Rana et al., 21 May 2026).

6. Sharp radii, extremals, and structural themes

Sharpness is not incidental in this literature; it is usually the central issue. Across analytic, harmonic, Cesàro, multivariable, and fractional settings, the admissible radius is typically the unique positive root of an explicit algebraic or transcendental equation. Representative examples include

$\mathbb D=\{z\in\mathbb C:|z|<1\}$78

and

$\mathbb D=\{z\in\mathbb C:|z|<1\}$79

The uniqueness of these roots is often established by monotonicity arguments, and the resulting constants are then shown to be best possible (Kumar et al., 2020, Ponnusamy et al., 2020, Ahamed et al., 2021, Mogbademu et al., 25 Sep 2025).

The extremal functions are equally recurrent. For bounded analytic classes, Möbius automorphisms such as

$\mathbb D=\{z\in\mathbb C:|z|<1\}$80

repeatedly witness sharpness. In harmonic fully starlike classes the extremal is

$\mathbb D=\{z\in\mathbb C:|z|<1\}$81

while in the univalent setting the Koebe function

$\mathbb D=\{z\in\mathbb C:|z|<1\}$82

remains extremal. Close-to-convex subclasses use explicit class-specific extremals such as

$\mathbb D=\{z\in\mathbb C:|z|<1\}$83

and octonionic theory uses slice regular Möbius-type maps built from the slice product and slice reciprocal (Ahamed et al., 2021, Allu et al., 2024, Ahamed et al., 11 Dec 2025, Ahammed et al., 26 Jun 2025, Rana et al., 21 May 2026).

A recurrent misconception is that the Bohr radius is always $\mathbb D=\{z\in\mathbb C:|z|<1\}$84. The collected results show otherwise. The radius can increase, as in the Cesàro, Libera, or Alexander settings; it can decrease, as in fractional derivative problems or close-to-convex subclasses; and it can depend explicitly on structural parameters such as $\mathbb D=\{z\in\mathbb C:|z|<1\}$85, $\mathbb D=\{z\in\mathbb C:|z|<1\}$86, $\mathbb D=\{z\in\mathbb C:|z|<1\}$87, $\mathbb D=\{z\in\mathbb C:|z|<1\}$88, $\mathbb D=\{z\in\mathbb C:|z|<1\}$89, $\mathbb D=\{z\in\mathbb C:|z|<1\}$90, $\mathbb D=\{z\in\mathbb C:|z|<1\}$91, or $\mathbb D=\{z\in\mathbb C:|z|<1\}$92 in the Riemann–Liouville order (Kumar et al., 2020, Zhou et al., 5 Feb 2025, Wang et al., 25 May 2026, Ahamed et al., 11 Dec 2025, Mogbademu et al., 25 Sep 2025).

The broader pattern is that Bohr-type inequalities are controlled by geometry, normalization, and the form of the majorant. In bounded analytic classes the right-hand side is often $\mathbb D=\{z\in\mathbb C:|z|<1\}$93; in geometric harmonic classes it is frequently the boundary distance

$\mathbb D=\{z\in\mathbb C:|z|<1\}$94

for the Cesàro operator it becomes

$\mathbb D=\{z\in\mathbb C:|z|<1\}$95

and in operator theory it is replaced by matrix-order or Jensen-type bounds. A plausible implication is that current Bohr theory is less a single theorem than a general program: identify a natural majorant, determine the largest radius on which it is dominated by the correct geometric or operator-theoretic benchmark, and prove that both the radius and the accompanying constants are sharp.