---
title: Bohr-Rogosinski Inequality
url: https://www.emergentmind.com/topics/bohr-rogosinski-inequality
type: topic
---

# Bohr-Rogosinski Inequality

The Bohr-Rogosinski inequality is a hybrid of the classical Bohr inequality and Rogosinski’s inequality for analytic or harmonic expansions on the unit disk. In its standard analytic form, if \(f(z)=\sum_{k=0}^{\infty} a_k z^k\) is analytic in \(\mathbb D\) and \(|f(z)|<1\), one studies the mixed majorant
\[
R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,
\]
and asks for the largest radius on which \(R_N(z)\le 1\). Subsequent work replaced the constant bound \(1\) by the Euclidean distance \(d(f(0),\partial f(\mathbb D))\), introduced squared, refined, and area-based variants, and extended the phenomenon to subordination classes, harmonic univalent mappings, concave and close-to-convex functions, Schwarz-function compositions, Banach-space holomorphic mappings, and operator-valued settings [1708.05585].

## 1. Classical analytic form

For bounded analytic functions on the unit disk, the classical background consists of Bohr’s inequality and Rogosinski’s inequality. If \(f(z)=\sum_{n=0}^{\infty} a_n z^n\) is analytic in \(\mathbb D\) and \(|f(z)|<1\), then
\[
\sum_{n=0}^{\infty}|a_n|r^n\le 1\quad \text{for } r\le \frac13,
\]
and the radius \(1/3\) is sharp. Rogosinski’s inequality controls partial sums \(S_N(z)=\sum_{k=0}^N a_k z^k\), with sharp radius \(1/2\). The Bohr-Rogosinski problem combines pointwise control of \(f\) with a tail sum of coefficients [1708.05585].

A decisive formulation was given by Kayumov and Ponnusamy: for \(N\ge 1\),
\[
|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k\le 1
\]
for \(r<R_N\), where \(R_N\) is the positive root of
\[
\Phi_N(r):=2(1+r)r^N-(1-r)^2=0.
\]
They also proved the squared variant
\[
|f(z)|^2+\sum_{k=N}^{\infty}|a_k|r^k\le 1
\]
for \(r<R_N'\), where \(R_N'\) is the positive root of
\[
(1+r)r^N-(1-r)^2=0.
\]
Both radii are sharp, and the same work treated the generalized form
\[
|f(z^m)|+\sum_{k=N}^{\infty}|a_k|r^k\le 1
\]
with \(R_{m,N}\) determined by
\[
2r^N(1+r^m)-(1-r)(1-r^m)=0
\]
[1708.05585].

A parallel analytic direction replaces coefficients or initial Taylor data by derivatives. For bounded analytic \(f\), the quantities
\[
A_f(z):=|f(z^m)|+|z^m||f'(z^m)|+\sum_{k=2}^{\infty}|a_k|r^k,
\]
\[
B_f(z):=|f(z^m)|+\sum_{k=2}^{\infty}\left|\frac{f^{(k)}(z^m)}{k!}\right|r^k,
\]
and
\[
C_f(z):=|f(z^m)|+|z|\,|f'(z^m)|+\sum_{k=2}^{\infty}|a_k|r^k
\]
satisfy sharp Bohr-Rogosinski-type bounds for radii given by explicit root equations \(\varphi_m(r)=0\), \(\psi_m(r)=0\), and \(\Phi_m(r)=0\), respectively [2004.08895]. These variants show that the inequality is not confined to the original tail-majorant form.

## 2. Distance form, subordination, and geometric classes

A major reformulation replaces the ambient bound \(1\) by a geometric distance term. If \(f\) is univalent in \(\mathbb D\), \(g(z)=\sum_{k=0}^{\infty} b_k z^k\), and \(g\prec f\), then with \(\Omega=f(\mathbb D)\) one has
\[
|g(z)|+\sum_{k=1}^{\infty}|b_k|r^k
 \le |f(0)|+\operatorname{dist}(f(0),\partial\Omega)
\]
for \(r<r_f\); the sharp univalent and convex-univalent radii were established in the same framework [1708.05585]. This distance form became the standard template for many later generalizations.

For concave univalent functions, the relevant class is \(Co(\alpha)\), \(\alpha\in[1,2]\), consisting of normalized univalent maps whose complements are convex and whose image at infinity has opening angle at most \(\pi\alpha\). If \(f\in Co(\alpha)\), \(g\in S(f)\), and \(\omega_0\) is a Schwarz function, then for each \(N\in\mathbb N\),
\[
|g(\omega_0(z))|+\sum_{n=N}^{\infty}|b_n|r^n
\le |f(0)|+d(f(0),\partial f(\mathbb D))
\]
holds for
\[
|z|=r<\min\{r_{\alpha,m_0},1/3\},
\]
where \(r_{\alpha,m_0}\) is the positive root in \((0,1)\) of
\[
F_{\alpha,m_0}(x):=\sum_{n=N}^{\infty}A_nx^n+f_\alpha(x^{m_0})-1=0.
\]
The radius is sharp, and the extremal function is
\[
f_\alpha(z)=2\left(\frac{1}{(1-z)^\alpha}-1\right)
\]
[2204.14085].

Recent work on subclasses of close-to-convex functions adapts the same distance formulation using sharp coefficient and distortion estimates. For the subclass \(\mathcal C_1\), for example,
\[
|f(z)|+\sum_{n=N}^{\infty}|a_n z^n|\le d(f(0),\partial f(\mathbb D))
\]
holds for \(|z|\le r_{1,N}\), where \(r_{1,N}\) is defined by an explicit root equation involving \(\log(1-r)\) and the coefficient majorant \(\sum_{n=N}^{\infty}\left(2-\frac1n\right)r^n\); analogous sharp results were obtained for \(\mathcal C_2\) and \(\mathcal C_3\) [2605.22930]. The common pattern is geometric: the coefficient tail is balanced against a class-dependent lower bound for the distance from \(f(0)\) to the image boundary.

## 3. Harmonic mappings and univalent harmonic classes

The harmonic theory starts with mappings \(f=h+\overline g\) on \(\mathbb D\), with \(h\) and \(g\) analytic. A central class is
\[
\mathcal P_{\mathcal H(M)}^0
=
\left\{
f=h+\overline g\in \mathcal H_0:
\operatorname{Re}(zh''(z))>-M+|zg''(z)|,\ z\in\mathbb D,\ M>0
\right\},
\]
where \(h(0)=0\), \(h'(0)=1\), \(g(0)=g'(0)=0\), and
\[
f(z)=z+\sum_{n=2}^{\infty} a_n z^n+\overline{\sum_{n=2}^{\infty} b_n z^n}.
\]
For this class one has the sharp coefficient estimate
\[
|a_n|+|b_n|\le \frac{2M}{n(n-1)},
\]
the growth bound
\[
|f(z)|\le |z|+2M\sum_{n=2}^{\infty}\frac{|z|^n}{n(n-1)},
\]
and
\[
d(f(0),\partial f(\mathbb D))=1+2M(1-2\ln 2)
\]
[2012.07837].

The sharp Bohr-Rogosinski inequality in this setting states that for \(f\in \mathcal P_{\mathcal H(M)}^0\) and any integer \(N\ge 2\),
\[
|f(z)|+\sum_{n=N}^{\infty} (|a_n|+|b_n|)|z|^n
\le d(f(0),\partial f(\mathbb D))
\]
for \(|z|=r\le r_N(M)\), where \(r_N(M)\) is the smallest root in \((0,1)\) of
\[
r-1+2M\left[2r-1+2(1-r)\ln(1-r)-\sum_{n=2}^{N-1}\frac{r^n}{n(n-1)}+\ln 4\right]=0.
\]
The corresponding squared version,
\[
|f(z)|^2+\sum_{n=N}^{\infty} (|a_n|+|b_n|)|z|^n
\le d(f(0),\partial f(\mathbb D)),
\]
is also sharp. Equality is attained for
\[
f_M(z)=z+2M\sum_{n=2}^{\infty}\frac{z^n}{n(n-1)}
\]
[2012.07837].

The same class is described as the class of fully starlike univalent functions for \(0<M<1/\log 4\). In that formulation, one obtains the sharp Bohr-Rogosinski-type inequality
\[
|z|+|f(z)|+\sum_{n=2}^{\infty}(|a_n|+|b_n|)|z|^n
\le d(f(0),\partial f(\mathbb D))
\]
for \(|z|=r\le r_M\), where \(r_M\) is the unique root in \((0,1)\) of
\[
2r+4M\big(r+(1-r)\log(1-r)\big)-1-2M(1-2\log 2)=0.
\]
The same paper also established harmonic variants involving the area \(S_r\), the Jacobian \(J_f\), and refined quadratic coefficient terms [2104.04509]. These results place the harmonic Bohr-Rogosinski phenomenon within geometric function theory rather than within bounded analytic function theory alone.

## 4. Refined, improved, and quasiconformal variants

A large modern branch of the subject introduces nonnegative correction terms that preserve sharpness while incorporating more geometric or energy-type information. For \(f(z)=\sum_{n=0}^{\infty} a_n z^n\in\mathcal B\), the area quantity
\[
S_r(f)=\int_{\mathbb D_r}|f'(z)|^2\,dA(z)=\pi\sum_{n=1}^{\infty}n|a_n|^2r^{2n}
\]
yields refined inequalities such as
\[
|a_0|+\sum_{n=1}^{\infty}|a_n|r^n+\frac{16}{9}\frac{S_r}{\pi}\le 1
\quad \text{for } r\le \frac13,
\]
and
\[
|a_0|^2+\sum_{n=1}^{\infty}|a_n|r^n+\frac{9}{8}\frac{S_r}{\pi}\le 1
\quad \text{for } r\le \frac12.
\]
More recent sharp Bohr-Rogosinski inequalities incorporate both \(S_r/\pi\) and \((S_r/\pi)^2\), together with
\[
A(f_0,r)=\left(\frac{1}{1+|a_0|}+\frac{r}{1-r}\right)\sum_{n=1}^{\infty}|a_n|^2r^{2n},
\]
at the radius \((\sqrt{17}-3)/4\) [2512.04768].

In harmonic settings with controlled dilatation, one studies \(K\)-quasiconformal sense-preserving harmonic mappings \(f=h+\overline g\), where \(|g'(z)/h'(z)|\le k<1\) and \(K=(1+k)/(1-k)\). For such mappings, sharp Bohr-Rogosinski inequalities relate \(|h(z)|\) or \(\operatorname{Re}h(z)\) to \(\sum (|a_n|+|b_n|)r^n\), with radii determined by explicit algebraic equations depending on \(K\) or \(k\). One representative form is
\[
|h(z)|+\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\le 1
\]
for \(r<r_1(k)\), where \(r_1(k)\) is the unique root of
\[
2(k+1)r(1+r)-(1-r)^2=0.
\]
Other sharp versions replace initial coefficients by \(|h(z)|^p\), \(|h'(z)|\), or add the area term
\[
S_\rho(h)=\int_{|z|<\rho}|h'(z)|^2\,dA
\]
[2312.15945; 2411.03352].

Subordination-based quasiconformal harmonic results likewise combine the Bohr-Rogosinski pattern with geometric target classes. If the analytic part is subordinate to a concave univalent function or to a Ma-Minda convex or starlike function, the radius is characterized as the unique root of a class-specific equation involving \(K\), canonical majorant functions, and distance-to-boundary estimates [2508.00012].

## 5. Several complex variables, Banach spaces, and operator-valued forms

The one-variable inequality has been lifted to several complex variables by replacing Taylor coefficients with homogeneous polynomials. If \(f:B_X\to \mathbb D\) is holomorphic on the unit ball of a complex Banach space \(X\) and
\[
f(z)=\sum_{s=0}^{\infty}P_s(z),
\]
then refined Bohr-Rogosinski inequalities control
\[
|f(v(z))|^p+\sum_{s=N}^{\infty}|P_s(z)|
\]
together with quadratic correction terms, where \(v:B_X\to B_X\) is a Schwarz mapping having a zero of order \(m\) at \(0\). The sharp radius \(R_{N,p,m}\) is the unique positive root of an explicit equation in \(r\) [2409.16610].

A closely related multivariable theory involves Schwarz functions explicitly. For bounded holomorphic functions in complete circular or convex Reinhardt domains, one proves inequalities of the form
\[
|f(\widehat\omega_{m_0}(z))|^p+\sum_{n=N}^{\infty}|P_n(\widehat\omega_k(z))|\le 1,
\]
where \(\widehat\omega_k\) vanishes to order \(k\) at the origin. In the multidimensional analogue, the sharp radius is the minimal root in \((0,1)\) of
\[
Y(r)=2r^{kN}(1+r^{m_0})-p(1-r^{m_0})(1-r^k)=0
\]
[2312.05635]. For the unit polydisc \(\mathbb D^n\), the sharp Bohr radius remains \(R_n=1/(3n)\), and sharp Bohr-Rogosinski radii were obtained for compositions with \(\omega_{n,m}\in\mathcal B_{n,m}\), as well as for inequalities involving the Euler operator
\[
Df(z)=\sum_{k=1}^{n} z_k \frac{\partial f(z)}{\partial z_k}
\]
[2601.06630].

Operator-valued and vector-valued versions replace scalar coefficients by bounded operators or Fréchet derivatives. For operator-valued holomorphic functions on simply connected domains, a general weighted Bohr-Rogosinski inequality uses a sequence \(\varphi=\{\varphi_n(r)\}\) of non-negative continuous functions and yields
\[
M_f(r):=\|f(\omega(z))\|^p\varphi_0(r)+\mu(r)\sum_{n=N}^{\infty}\|A_n\|\varphi_n(r)\le \varphi_0(r),
\]
with sharp radius determined by the minimal positive root of the corresponding weight equation [2411.04000]. Operator-valued analogues of multidimensional refined and improved Bohr-Rogosinski inequalities, including terms involving \(S_r/\pi\), were established for complete circular domains [2308.13757]. Vector-valued holomorphic functions with lacunary series on finite-dimensional Banach sequence spaces admit sharp Bohr-Rogosinski inequalities in terms of Fréchet derivatives and Schwarz mappings of prescribed order [2509.03532].

## 6. Sharpness, extremals, and contemporary generalizations

Sharpness is structural rather than incidental in this theory. In the classical disk setting, extremality is often realized by Möbius maps \(f_a(z)=\frac{a-z}{1-az}\); in subordination problems it is tied to the Koebe function or convex extremals; in concave classes it is realized by
\[
f_\alpha(z)=2\left(\frac{1}{(1-z)^\alpha}-1\right);
\]
and in the harmonic class \(\mathcal P_{\mathcal H(M)}^0\) it is realized by
\[
f_M(z)=z+2M\sum_{n=2}^{\infty}\frac{z^n}{n(n-1)}.
\]
Operator-valued and Banach-space versions likewise use explicit Blaschke-type, Möbius-type, or sliced extremal functions to show that the radii cannot be increased [1708.05585; 2204.14085; 2012.07837; 2409.16610].

The current literature also treats the Bohr-Rogosinski inequality as part of broader radius problems for differential-inequality classes of harmonic mappings. For the generalized class \(\mathcal{BH}_0(\gamma,\delta)\), one has
\[
\left|f(z^m)\right|+\sum_{n=N}^{\infty}(|a_n|+|b_n|)|z|^n
\le d(f(0),\partial f(\mathbb D))
\]
with sharp radius \(R_{m,N}(\gamma,\delta)\) given by a unique root of an explicit equation, together with area-term and higher-order coefficient-sum refinements [2606.02612]. For the close-to-convex harmonic class \(R_H^0(\gamma,\delta,\lambda)\), the sharp inequality
\[
|f(z^n)|+\sum_{m=N}^{\infty}(|a_m|+|b_m|)|z|^m
\le d(f(0),\partial f(\mathbb D))
\]
holds up to a radius \(R_{n,N}(\gamma,\delta,\lambda)\) defined by an explicit root condition, and a further refined version includes quadratic coefficient terms [2605.14776]. This suggests that the modern subject treats the Bohr-Rogosinski inequality as a family of sharp radius problems governed by coefficient estimates, growth theorems, geometric distance bounds, and extremal mappings.

Across these settings, the invariant core is unchanged: a pointwise term such as \(|f(z)|\), \(|f(z)|^2\), \(|f(v(z))|^p\), or \(|f(z^m)|\) is coupled to a coefficient tail, and the optimal radius is determined by the boundary between local coefficient control and global image geometry.

Source: https://www.emergentmind.com/topics/bohr-rogosinski-inequality