Bohr-Rogosinski Inequality

• Bohr-Rogosinski Inequality is a hybrid inequality that extends classical Bohr and Rogosinski bounds to analytic and harmonic mappings on the unit disk.
• It employs mixed majorants combining the function value and tail coefficients, with sharp radii determined by explicit root equations.
• Extensions include distance formulations, harmonic and operator-valued variants, underscoring its wide applicability in geometric function theory.

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 (Kayumov et al., 2017).

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

$\mathbb D$0

and the radius $\mathbb D$1 is sharp. Rogosinski’s inequality controls partial sums $\mathbb D$2, with sharp radius $\mathbb D$3. The Bohr-Rogosinski problem combines pointwise control of $\mathbb D$4 with a tail sum of coefficients (Kayumov et al., 2017).

A decisive formulation was given by Kayumov and Ponnusamy: for $\mathbb D$5,

$\mathbb D$6

for $\mathbb D$7, where $\mathbb D$8 is the positive root of

$\mathbb D$9

They also proved the squared variant

$|f(z)|<1$0

for $|f(z)|<1$1, where $|f(z)|<1$2 is the positive root of

$|f(z)|<1$3

Both radii are sharp, and the same work treated the generalized form

$|f(z)|<1$4

with $|f(z)|<1$5 determined by

$|f(z)|<1$6

(Kayumov et al., 2017).

A parallel analytic direction replaces coefficients or initial Taylor data by derivatives. For bounded analytic $|f(z)|<1$7, the quantities

$|f(z)|<1$8

$|f(z)|<1$9

and

$$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$0

satisfy sharp Bohr-Rogosinski-type bounds for radii given by explicit root equations $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$1, $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$2, and $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$3, respectively (Alkhaleefah et al., 2020). 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 $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$4 by a geometric distance term. If $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$5 is univalent in $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$6, $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$7, and $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$8, then with $$R_N(z):=|f(z)|+\sum_{k=N}^{\infty}|a_k|r^k,\qquad |z|=r,$$9 one has

$R_N(z)\le 1$0

for $R_N(z)\le 1$1; the sharp univalent and convex-univalent radii were established in the same framework (Kayumov et al., 2017). This distance form became the standard template for many later generalizations.

For concave univalent functions, the relevant class is $R_N(z)\le 1$2, $R_N(z)\le 1$3, consisting of normalized univalent maps whose complements are convex and whose image at infinity has opening angle at most $R_N(z)\le 1$4. If $R_N(z)\le 1$5, $R_N(z)\le 1$6, and $R_N(z)\le 1$7 is a Schwarz function, then for each $R_N(z)\le 1$8,

$R_N(z)\le 1$9

holds for

$1$0

where $1$1 is the positive root in $1$2 of

$1$3

The radius is sharp, and the extremal function is

$1$4

(Allu et al., 2022).

Recent work on subclasses of close-to-convex functions adapts the same distance formulation using sharp coefficient and distortion estimates. For the subclass $1$5, for example,

$1$6

holds for $1$7, where $1$8 is defined by an explicit root equation involving $1$9 and the coefficient majorant $d(f(0),\partial f(\mathbb D))$0; analogous sharp results were obtained for $d(f(0),\partial f(\mathbb D))$1 and $d(f(0),\partial f(\mathbb D))$2 (Rana et al., 21 May 2026). The common pattern is geometric: the coefficient tail is balanced against a class-dependent lower bound for the distance from $d(f(0),\partial f(\mathbb D))$3 to the image boundary.

3. Harmonic mappings and univalent harmonic classes

The harmonic theory starts with mappings $d(f(0),\partial f(\mathbb D))$4 on $d(f(0),\partial f(\mathbb D))$5, with $d(f(0),\partial f(\mathbb D))$6 and $d(f(0),\partial f(\mathbb D))$7 analytic. A central class is

$d(f(0),\partial f(\mathbb D))$8

where $d(f(0),\partial f(\mathbb D))$9, $f(z)=\sum_{n=0}^{\infty} a_n z^n$0, $f(z)=\sum_{n=0}^{\infty} a_n z^n$1, and

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

For this class one has the sharp coefficient estimate

$f(z)=\sum_{n=0}^{\infty} a_n z^n$3

the growth bound

$f(z)=\sum_{n=0}^{\infty} a_n z^n$4

and

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

(Ahamed et al., 2020).

The sharp Bohr-Rogosinski inequality in this setting states that for $f(z)=\sum_{n=0}^{\infty} a_n z^n$6 and any integer $f(z)=\sum_{n=0}^{\infty} a_n z^n$7,

$f(z)=\sum_{n=0}^{\infty} a_n z^n$8

for $f(z)=\sum_{n=0}^{\infty} a_n z^n$9, where $\mathbb D$0 is the smallest root in $\mathbb D$1 of

$\mathbb D$2

The corresponding squared version,

$\mathbb D$3

is also sharp. Equality is attained for

$\mathbb D$4

(Ahamed et al., 2020).

The same class is described as the class of fully starlike univalent functions for $\mathbb D$5. In that formulation, one obtains the sharp Bohr-Rogosinski-type inequality

$\mathbb D$6

for $\mathbb D$7, where $\mathbb D$8 is the unique root in $\mathbb D$9 of

$|f(z)|<1$0

The same paper also established harmonic variants involving the area $|f(z)|<1$1, the Jacobian $|f(z)|<1$2, and refined quadratic coefficient terms (Ahamed et al., 2021). 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)|<1$3, the area quantity

$|f(z)|<1$4

yields refined inequalities such as

$|f(z)|<1$5

and

$|f(z)|<1$6

More recent sharp Bohr-Rogosinski inequalities incorporate both $|f(z)|<1$7 and $|f(z)|<1$8, together with

$|f(z)|<1$9

at the radius $\mathbb D$00 (Aahmed et al., 4 Dec 2025).

In harmonic settings with controlled dilatation, one studies $\mathbb D$01-quasiconformal sense-preserving harmonic mappings $\mathbb D$02, where $\mathbb D$03 and $\mathbb D$04. For such mappings, sharp Bohr-Rogosinski inequalities relate $\mathbb D$05 or $\mathbb D$06 to $\mathbb D$07, with radii determined by explicit algebraic equations depending on $\mathbb D$08 or $\mathbb D$09. One representative form is

$\mathbb D$10

for $\mathbb D$11, where $\mathbb D$12 is the unique root of

$\mathbb D$13

Other sharp versions replace initial coefficients by $\mathbb D$14, $\mathbb D$15, or add the area term

$\mathbb D$16

(Ahamed et al., 2023, Biswas et al., 2024).

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 $\mathbb D$17, canonical majorant functions, and distance-to-boundary estimates (Ahamed et al., 21 Jul 2025).

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 $\mathbb D$18 is holomorphic on the unit ball of a complex Banach space $\mathbb D$19 and

$\mathbb D$20

then refined Bohr-Rogosinski inequalities control

$\mathbb D$21

together with quadratic correction terms, where $\mathbb D$22 is a Schwarz mapping having a zero of order $\mathbb D$23 at $\mathbb D$24. The sharp radius $\mathbb D$25 is the unique positive root of an explicit equation in $\mathbb D$26 (Ahamed et al., 2024).

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

$\mathbb D$27

where $\mathbb D$28 vanishes to order $\mathbb D$29 at the origin. In the multidimensional analogue, the sharp radius is the minimal root in $\mathbb D$30 of

$\mathbb D$31

(Ahammed et al., 2023). For the unit polydisc $\mathbb D$32, the sharp Bohr radius remains $\mathbb D$33, and sharp Bohr-Rogosinski radii were obtained for compositions with $\mathbb D$34, as well as for inequalities involving the Euler operator

$\mathbb D$35

(Ahamed et al., 10 Jan 2026).

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 $\mathbb D$36 of non-negative continuous functions and yields

$\mathbb D$37

with sharp radius determined by the minimal positive root of the corresponding weight equation (Ahammed et al., 2024). Operator-valued analogues of multidimensional refined and improved Bohr-Rogosinski inequalities, including terms involving $\mathbb D$38, were established for complete circular domains (Ahammed et al., 2023). 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 (Ahammed et al., 26 Aug 2025).

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 $\mathbb D$39; in subordination problems it is tied to the Koebe function or convex extremals; in concave classes it is realized by

$\mathbb D$40

and in the harmonic class $\mathbb D$41 it is realized by

$\mathbb D$42

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 (Kayumov et al., 2017, Allu et al., 2022, Ahamed et al., 2020, Ahamed et al., 2024).

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 $\mathbb D$43, one has

$\mathbb D$44

with sharp radius $\mathbb D$45 given by a unique root of an explicit equation, together with area-term and higher-order coefficient-sum refinements (Wang et al., 25 May 2026). For the close-to-convex harmonic class $\mathbb D$46, the sharp inequality

$\mathbb D$47

holds up to a radius $\mathbb D$48 defined by an explicit root condition, and a further refined version includes quadratic coefficient terms (Kumar, 14 May 2026). 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 $\mathbb D$49, $\mathbb D$50, $\mathbb D$51, or $\mathbb D$52 is coupled to a coefficient tail, and the optimal radius is determined by the boundary between local coefficient control and global image geometry.