---
title: Bohr-Type Inequalities in Complex Analysis
url: https://www.emergentmind.com/topics/bohr-type-inequalities
type: topic
---

# Bohr-Type Inequalities in Complex Analysis

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 [1911.05315, 2104.04509, 1107.1289].

## 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 \(1-|a_0|\) by \(1-|a_0|^2\) gives radius \(1/2\), while imposing \(a_0=0\) gives sharp radius \(1/\sqrt2\) [1911.05315].

A separate but historically related usage appears in operator theory. There the scalar inequality
\[
|a+b|^2 \le p|a|^2+q|b|^2,\qquad p,q>0,\quad \frac1p+\frac1q=1,
\]
is treated as the classical Bohr inequality, and it admits operator, matrix-order, Jensen-type, and eigenvalue generalizations. This branch includes inequalities such as
\[
\left|\sum_{i=1}^n t_i A_i\right|^2 \le \sum_{i=1}^n t_i |A_i|^2,
\qquad t_i>0,\ \sum_i t_i=1,
\]
which the literature interprets as an operator Jensen inequality for \(K(z)=|z|^2\) [1107.1289].

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 \(f\in\mathcal B\),
\[
\sum_{n=0}^\infty |a_n|\,r^n \le \frac{1-r|f|_2}{1-r}, \qquad r\in[0,1),
\]
where \(|f|_2=\sum_{n=0}^\infty |a_n|^2 r^{2n}\). Further sharp inequalities of the form
\[
\sum_{n=1}^\infty |a_n|\,r^n + \frac{1}{1+|a_1|}\sum_{n=1}^\infty |a_n|^2 r^{2n} <1
\]
hold for \(r<3/5\), and related variants hold for \(r<(\sqrt5-1)/2\), \(r<1/2\), or \(r<1/(2+|a_0|)\), depending on the normalization and the quadratic term used [1911.05315].

Another direction adds area-type corrections. For bounded analytic functions, one has sharp inequalities such as
\[
|f(z)|+\sum_{k=1}^\infty |a_k|r^k+2(\sqrt5-1)\frac{S_r}{\pi}\le 1
\qquad (|z|=r<\sqrt5-2),
\]
where \(S_r\) is the Euclidean area of \(f(\mathbb D_r)\). The same paper also recalls the sharp earlier estimate
\[
|f(z)|^2+\sum_{k=1}^\infty |a_k|r^k\le 1
\qquad (|z|=r\le 1/3),
\]
showing that replacing \(|f(z)|\) by \(|f(z)|^2\) restores the classical radius \(1/3\) [2004.08625].

Area refinements became more delicate in later work. For instance, a stronger nonlinear correction replaces \(S_r/\pi\) by
\[
\frac{S_r}{\pi-S_r},
\]
leading to sharp inequalities such as
\[
\sum_{n=0}^\infty |a_n|r^n+\frac{S_r}{\pi-S_r}\le 1 \qquad (r\le 1/3),
\]
and
\[
\frac{|a_0|}{2}+\sum_{n=1}^\infty |a_n|r^n+\frac{S_r}{\pi-S_r}\le 1
\qquad (r\le 1/2).
\]
The same line of work also proves sharper parameterized versions involving \(\frac{1+|a_0|}{1-|a_0|}r\), \(|f(z)-a_0|\), and explicit sharp constants \(\lambda\) determined by algebraic equations [2312.15945].

Vanishing order at the origin produces another class of refinements. For
\[
\mathcal B_k=\{ f\in\mathcal B : f(0)=f'(0)=\cdots=f^{(k-1)}(0)=0\},
\]
sharp inequalities include
\[
\sum_{n=k}^\infty |a_n|\, r^n
+\frac{1+r}{1-r}\sum_{n=k+1}^\infty |a_n|^2\, r^{2n-k}
\le 1
\]
for \(r\le R_k\), where \(R_k\) is the unique root of
\[
4(1-r)-r^k(1-2r+5r^2)=0.
\]
A second sharp variant uses
\[
\sum_{n=k}^\infty |a_n|\,r^n
+\frac{r^k}{1+r}\sum_{n=k+1}^\infty |a_n|^2\,r^{2n-k}
\le 1
\]
for \(r\le S_k\), where \(S_k\) is the unique solution of
\[
2(1-r)-r^k(3-r)=0.
\]
These results show that prescribed multiple zeros improve the admissible radius in a structured way [2006.06441].

## 3. Weighted, parameterized, and operator-transform formulations

A modern branch of the subject replaces the monomial weights \(r^n\) by more general operator- or weight-dependent terms. For the Cesàro operator,
\[
\mathcal C f(z)=\sum_{n=0}^\infty \frac{1}{n+1}\left(\sum_{k=0}^n a_k\right)z^n
=\int_0^1 \frac{f(tz)}{1-tz}\,dt,
\]
the associated Bohr sum is
\[
\mathcal C_f(r)=\sum_{n=0}^\infty \frac{1}{n+1}\left(\sum_{k=0}^n |a_k|\right)r^n.
\]
If \(f\in\mathcal B\), then
\[
\mathcal C_f(r)\le \frac1r\log\frac1{1-r}
\qquad (r<R),
\]
where \(R\approx 0.5335\) is the positive root of
\[
2x-3(1-x)\log\frac1{1-x}=0.
\]
This is sharp. The same paper gives sharp radii for the \(\beta\)-Cesàro, Bernardi, Libera, and Alexander operators, with the Libera/Alexander radius
\[
R\approx 0.5828
\]
defined by
\[
3x+2\log(1-x)=0
\]
[2008.00468].

The Cesàro theory has also been refined by inserting derivative data. If \(f\in\mathcal B\), then
\[
|\mathcal C f(z)| +|\mathcal C f'(z)|\,\phi_1(r) +\sum_{k=2}^\infty |a_k|\phi_k(r)
\le \frac1r\log\frac1{1-r}
\]
for \(|z|=r\le r_1\le R\approx 0.493411\), where \(r_1\) 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
\[
f(z)=\frac{a+z}{1+az}
\]
with \(a\to1^{-}\) [2411.01437].

Weighted and parameterized frameworks generalize the majorant itself. One approach introduces two parameters \(\alpha,\beta\) and proves inequalities such as
\[
\alpha |f(z)|+(1-\alpha)a+\beta\sum_{k=1}^\infty |a_k|\,|z|^k\le 1
\]
for \(|z|=r<R_1\), where \(R_1\) is the positive root of
\[
(1+2\beta-2\alpha)r^2+(2\alpha+2\beta)r-1=0.
\]
Another introduces parameters \(p,\lambda\) and proves
\[
|f(z)|^p+\lambda \sum_{k=1}^\infty |a_k|\,|z|^k \le 1
\qquad (|z|=r<R_{\lambda,p}),
\]
with
\[
R_{\lambda,p}= \begin{cases}
\dfrac{-p-\lambda+\sqrt{\lambda^2+4p\lambda}}{2\lambda-p}, & p\ne 2\lambda,\\[1.2ex]
\dfrac13, & p=2\lambda.
\end{cases}
\]
These families recover classical and one-parameter Bohr inequalities as special cases [2502.02824, 2502.02828].

A still more flexible generalization replaces \(\{r^n\}\) by a weight sequence \(\varphi=\{\varphi_n(r)\}_{n=0}^\infty\in F\), where
\[
\Phi_N(r)=\sum_{n=N}^\infty \varphi_n(r).
\]
In this setting the papers study inequalities for \(f\circ w\), where \(w\in B_m\) is a Schwarz function with \(m\)-fold zero, and derive sharp radii from conditions such as \(\Upsilon_1(r)=0\), \(\Upsilon_2(r)=0\), and \(\Upsilon_5(r)=0\). This single-parameter framework with \(p\in(0,2]\) subsumes several earlier results when \(\varphi_n(r)=r^n\) [2302.07745].

## 4. Harmonic, quasiconformal, and multiple-zero settings

For harmonic mappings \(f=h+\overline g\) in \(\mathbb D\), Bohr-type theory must account for the coupled analytic and co-analytic coefficients. A central class is
\[
\mathcal P^0_{\mathcal H}(M)
=
\left\{
f=h+\overline g\in \mathcal H_0:
\operatorname{Re}(zh''(z))>-M+|zg''(z)|,\ z\in\mathbb D,\ M>0
\right\},
\]
with normalized series
\[
f(z)=z+\sum_{n=2}^{\infty}a_n z^n+\overline{\sum_{n=2}^{\infty}b_n z^n}.
\]
For \(0<M<1/\log 4\), functions in this class are fully starlike; the sharper range
\[
0<M<\frac{1}{2(\ln 4-1)}
\]
ensures positivity of
\[
1+2M(1-2\ln 2)>0.
\]
The coefficient theory is explicit:
\[
|a_n|+|b_n|\le \frac{2M}{n(n-1)},\qquad
\big||a_n|-|b_n|\big|\le \frac{2M}{n(n-1)},\qquad
|a_n|\le \frac{2M}{n(n-1)},
\]
with extremal function
\[
f_M(z)=z+2M\sum_{n=2}^{\infty}\frac{z^n}{n(n-1)}.
\]
The associated boundary-distance estimate is
\[
d\bigl(f(0),\partial f(\mathbb D)\bigr)\ge 1+2M(1-2\log 2),
\]
and equality is attained by \(f_M\) [2104.04509].

On this class one obtains several sharp Bohr-type inequalities. The Bohr–Rogosinski form is
\[
|z|+|f(z)|+\sum_{n=2}^{\infty}(|a_n|+|b_n|)|z|^n
\le d\bigl(f(0),\partial f(\mathbb D)\bigr)
\]
for \(r\le r_M\), where \(r_M\) is the unique root of
\[
2r+4M\bigl(r+(1-r)\log(1-r)\bigr)-1-2M(1-2\log 2)=0.
\]
There are also sharp refinements involving the area
\[
|f(z)|+\sum_{n=2}^{\infty}(|a_n|+|b_n|)|z|^n+\frac{S_r}{\pi}
\le d\bigl(f(0),\partial f(\mathbb D)\bigr),
\]
the Jacobian
\[
\sum_{n=2}^{\infty}(|a_n|+|b_n|)|z|^n+\sqrt{|J_f(z)|}\,|z|
\le d\bigl(f(0),\partial f(\mathbb D)\bigr),
\]
and a square-majorant form involving
\[
|f(z)|^2+\sum_{n=2}^{\infty}(|a_n|+|b_n|)|z|^n
+\frac{r}{1-r}\sum_{n=2}^{\infty}(|a_n|+|b_n|)^2|z|^{2n}.
\]
Each radius is defined as the unique root of an explicit equation and is best possible [2104.04509].

A more general two-parameter harmonic class is
\[
\mathcal{BH}^{0}(\gamma,\delta),
\]
defined by the second-order differential inequality
\[
\operatorname{Re}\!\left[\gamma \frac{h(z)}{z}+\delta h'(z)+\frac{\delta-\gamma}{2}zh''(z)\right]
>
\left|\gamma \frac{g(z)}{z}+\delta g'(z)+\frac{\delta-\gamma}{2}zg''(z)\right|,
\qquad \delta\ge \gamma\ge 0.
\]
Here the coefficient bounds are
\[
|a_n|+|b_n|\le
\frac{4(\delta+\gamma)}{(n+1)\,[n(\delta-\gamma)+2\gamma]},
\]
and one sharp improved Bohr inequality is
\[
|z|+\sum_{n=2}^{\infty}(|a_n|+|b_n|)|z|^n
+\sum_{n=2}^{\infty}(|a_n|+|b_n|)^p|z|^{pn}
\le d(f(0),\partial f(\mathbb D)).
\]
For \((\gamma,\delta)=(1,3)\), the paper reports
\[
r_2(1,3)\approx 0.309260,\qquad r_2^*(1,3)\approx 0.399130,\qquad
r^*(1,3)\approx 0.313516,
\]
and area-refined radii
\[
\dot r(1,3)\approx0.268346,\qquad \ddot r(1,3)\approx0.359414,
\]
all best possible [2606.02612].

Multiple zeros at the origin generate another harmonic branch. For \(p\)-symmetric harmonic mappings with
\[
h(z)=\sum_{n=k}^\infty a_{pn+m}z^{pn+m},\qquad
g(z)=\sum_{n=k}^\infty b_{pn+m}z^{pn+m},
\]
the literature proves sharp Bohr-type inequalities under either boundedness of both parts or the weaker domination
\[
|g'(z)|\le d|h'(z)|
\qquad (d\in[0,1]).
\]
Representative sharp radii are defined by equations such as
\[
r^{2(p-m)}-(8+4d)r^{p-m}+4(1+d)(3+d)r^{2p}+4=0,
\]
\[
2(1-r^p)-r^{pk+m}(3-r^p)=0,
\]
and
\[
(1+a)(1-r^p)-2r^{pk+m}\bigl(2a+a+r^p(1-2a)\bigr)=0,
\]
with extremals of the form
\[
f(z)=h(z)+\lambda h(z),\qquad
h(z)=z^{pk+m}\frac{a-z^p}{1-az^p}
\]
or \(h(z)=z^{pk+m}\) [2103.09403].

For \(K\)-quasiconformal harmonic mappings, recent work introduces multiple Schwarz functions \(\omega_p,\omega_m,\omega_q\) to majorize different terms independently. One sharp result states that, for suitable \(f=h+\overline g\),
\[
\sum_{n=0}^\infty |a_n|\,|\omega_p(z)|^n
+\sum_{n=2}^\infty |b_n|\,|\omega_m(z)|^n
\le \|h\|_\infty
\]
for \(r\le r_{p,m,k}\), where \(k=(K-1)/(K+1)\) and \(r_{p,m,k}\) is the smallest positive root of
\[
\frac{2r^p}{1-r^p}
+2k\left(\frac{r^m}{1-r^m}+\log(1-r^m)\right)-1=0.
\]
The same framework yields improved inequalities involving \(h(\omega_m)\), \(h'(\omega_m)\), and square/area-type corrections [2510.00684].

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

In several complex variables, refined Bohr theory on the polydisk \(\mathbb D^n\) uses the scaling parameter \(n\mathbf r\), where \(\mathbf r=\|z\|_\infty\). If
\[
f(z)=\sum_{|\alpha|=0}^{\infty} a_\alpha z^\alpha
\]
is holomorphic in the unit polydisk and \(|f(z)|\le 1\) on \(\mathbb P\Delta(0;1/n)\), then there are sharp multidimensional analogues of refined Bohr–Rogosinski inequalities, \(|f|^2\)-Bohr inequalities, and derivative-based results. The radial derivative is
\[
Df(z):=\sum_{k=1}^n z_k\frac{\partial f(z)}{\partial z_k},
\]
and one sharp multidimensional refinement is
\[
|f(z)|+|Df(z)|
+\sum_{k=2}^{\infty}\sum_{|\alpha|=k}|a_\alpha|\, r^\alpha
+\left( \frac{1}{1+|a_0|}+\frac{\mathbf r}{1-\mathbf r} \right)
\sum_{k=1}^{\infty}\sum_{|\alpha|=k}|a_\alpha|^2 r^{2\alpha}
\le 1
\]
for
\[
n\mathbf r\le \frac{\sqrt{17}-3}{4},
\]
and this constant is best possible. Higher-order mixed partials satisfy analogous sharp inequalities with radii determined by
\[
(1+n\mathbf r)(1-2n\mathbf r)(1-n\mathbf r)^{N-1}-2(n\mathbf r)^N=0
\]
or
\[
(1+n\mathbf r)(1-2n\mathbf r)(1-n\mathbf r)^{N-1}-(n\mathbf r)^N=0
\]
[2512.15752].

For holomorphic functions on the unit ball \(B_X\) of a finite-dimensional Banach sequence space, the theory extends to lacunary and alternating series. If
\[
f(z)=\sum_{k=0}^\infty \frac{D^{kp+m}f(0)(z^{kp+m})}{(kp+m)!}\in H(B_X,\mathbb D),
\]
then a sharp lacunary Bohr inequality holds for \(0<|z|=r<r_{p,m}\), where \(r_{p,m}\) is the unique root of
\[
G(r)=5r^{2p+m}-2r^{p+m}+pr^m+4r^p-4=0.
\]
In the special case \(m=0\),
\[
r_{p,0}=\left(\frac35\right)^{1/p}.
\]
The same paper proves vector-valued and alternating analogues with sharp radii given by
\[
r^{p+m}+r^{2p}-1=0
\qquad\text{and}\qquad
r^{2p+m}+2r^{2p}-1=0
\]
[2404.18623].

Slice regular function theory over the octonions provides a non-associative analogue. For
\[
f(x)=\sum_{k=0}^{\infty} x^k a_k,\qquad a_k\in\mathbb O,
\]
defined on the unit ball \(\mathbb B\subset\mathbb O\), the generalized Bohr inequality reads
\[
|a_0|^m+\sum_{k=1}^{\infty}|x^k a_k|\le 1
\qquad \text{whenever}\qquad |x|\le R_m:=\frac{m}{2+m},
\]
for every \(m\in(0,2]\), and \(R_m\) is best possible. Refined versions add terms such as \(|f(x)-a_0|^2\), square-sums of coefficients, or
\[
S_x^*:=\sum_{k=1}^{\infty} k\,|x^k a_k|^2.
\]
In the half-space case \(\Pi=\{x\in\mathbb O:\operatorname{Re}(x)\le 1\}\), the paper proves a sharp radius
\[
R_*\approx 0.24683,
\]
defined by
\[
3r^3-5r^2-3r+1=0
\]
[2507.01981].

Fractional calculus yields another branch. For the Riemann–Liouville derivative
\[
D^\alpha f(z)=\sum_{n=0}^{\infty} a_n \frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)} z^{n-\alpha},
\qquad 0<\alpha<1,
\]
the fractional Bohr sum is
\[
B[D^\alpha f]
=
\sum_{n=0}^{\infty}\frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)}|a_n|r^{n-\alpha}.
\]
For \(f\in\mathcal B\), one obtains a sharp radius \(R(\alpha)\) determined by
\[
\sum_{n=1}^\infty \frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)} r^n
= \frac{1}{2\Gamma(1-\alpha)}.
\]
The numerical values reported in the paper are
\[
\alpha: 0,\ 0.2,\ 0.5,\ 0.8,\ 0.99,\qquad
R(\alpha): 0.33333,\ 0.30841,\ 0.28301,\ 0.27026,\ 0.26796.
\]
For the \(|a_0|^2\)-weighted variant the sharp fractional radius \(N(\alpha)\) satisfies
\[
\alpha: 0,\ 0.1,\ 0.2,\ 0.5,\ 0.8,\ 0.9,\qquad
N(\alpha): 0.50000,\ 0.467028,\ 0.431574,\ 0.308621,\ 0.150656,\ 0.083639.
\]
The paper also gives univalent, convex, and Bloch-function versions, all with \(\alpha\)-dependent sharp radii [2509.20660].

A geometric subclass approach appears for close-to-convex functions. For the Silverman–Telage classes \(\mathcal C_1,\mathcal C_2,\mathcal C_3\), the literature proves sharp inequalities involving \(|f(z)|\), \(|f'(z)||z|\), coefficient tails, and \(p\)-power sums. Representative examples are
\[
|f(z)|+|f'(z)||z|+\sum_{n=2}^\infty |a_n z^n|
\le d\bigl(f(0),\partial f(\mathbb D)\bigr)
\]
for \(|z|\le r_{11}\), where
\[
r_{11}=0.110377
\]
is the unique solution of an explicit logarithmic equation in \(\mathcal C_1\), and
\[
r_{21}\simeq 0.173417
\]
defined by
\[
1-6r+r^2+2r^3=0
\]
in \(\mathcal C_2\) [2605.22930].

## 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
\[
2x-3(1-x)\log\frac1{1-x}=0,
\qquad
4(1-r)-r^k(1-2r+5r^2)=0,
\qquad
2r+4M\bigl(r+(1-r)\log(1-r)\bigr)-1-2M(1-2\log 2)=0,
\]
and
\[
\sum_{n=1}^\infty \frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)} r^n
= \frac{1}{2\Gamma(1-\alpha)}.
\]
The uniqueness of these roots is often established by monotonicity arguments, and the resulting constants are then shown to be best possible [2008.00468, 2006.06441, 2104.04509, 2509.20660].

The extremal functions are equally recurrent. For bounded analytic classes, Möbius automorphisms such as
\[
f_a(z)=\frac{a-z}{1-az}
\qquad\text{or}\qquad
f(z)=\frac{a+z}{1+az}
\]
repeatedly witness sharpness. In harmonic fully starlike classes the extremal is
\[
f_M(z)=z+2M\sum_{n=2}^{\infty}\frac{z^n}{n(n-1)},
\]
while in the univalent setting the Koebe function
\[
f(z)=\frac{z}{(1-z)^2}
\]
remains extremal. Close-to-convex subclasses use explicit class-specific extremals such as
\[
f(z)=\frac{z}{1-z}
\quad\text{or}\quad
f(z)=\frac{2z}{1+z}-\log(1+z),
\]
and octonionic theory uses slice regular Möbius-type maps built from the slice product and slice reciprocal [2104.04509, 2411.01437, 2512.15752, 2507.01981, 2605.22930].

A recurrent misconception is that the Bohr radius is always \(1/3\). 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 \(M\), \((\gamma,\delta)\), \((\alpha,\beta)\), \(m\), \(p\), \(K\), \(n\), or \(\alpha\) in the Riemann–Liouville order [2008.00468, 2502.02824, 2606.02612, 2512.15752, 2509.20660].

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 \(1\); in geometric harmonic classes it is frequently the boundary distance
\[
d\bigl(f(0),\partial f(\mathbb D)\bigr);
\]
for the Cesàro operator it becomes
\[
\frac1r\log\frac1{1-r};
\]
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.

Source: https://www.emergentmind.com/topics/bohr-type-inequalities