Bohr Inequality & Lacunary Series

• The topic presents Bohr inequalities for analytic functions with arithmetic-progression lacunarity, achieving enhanced admissible radii beyond the classical value of 1/3.
• Methodologies use Fréchet expansions and Banach-space frameworks to extend analysis to vector-valued holomorphic mappings and mixed Bohr–Rogosinski inequalities.
• Sharp radii are derived from explicit algebraic equations, offering practical insights for norm estimates and advanced lacunary functionals in various settings.

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to=shell 下载彩神争霸 code python - <<'PY' import requests, bs4, urllib.parse, re q=urllib.parse.quote('Bohr inequality lacunary series') url='https://arxiv.org/search/?query=%s&searchtype=all&abstracts=show&order=-announced_date_first&size=10' % q html=requests.get(url,headers={'User-Agent':'Mozilla/5.0'},timeout=20).text print('len',len(html)) soup=bs4.BeautifulSoup(html,'html.parser') results=soup.select('li.arxiv-result') for i,res in enumerate(results[:10],1): title=res.select_one('p.title') pid=res.select_one('p.list-title a') if title and pid: print(i, pid.text.strip(), '-', ' '.join(title.text.split())[:120]) PY

Bohr inequality with lacunary series concerns Bohr-type majorant estimates for analytic or holomorphic functions whose nonzero coefficients occur only on a sparse set of degrees, most commonly an arithmetic progression such as $\{kp+m:k\ge 0\}$ or $\{qs+m:s\ge 1\}$. In the classical one-variable form, if $f(z)=\sum_{n\ge 0}a_n z^n$ is analytic on $\mathbb D$ and $\|f\|_\infty\le 1$, then $\sum_{n\ge 0}|a_n|r^n\le 1$ for $0\le r\le 1/3$, and $1/3$ is sharp. The lacunary variant asks how this radius changes when many degrees are excluded. Recent work shows that arithmetic-progression lacunarity frequently enlarges the admissible radius, and that the phenomenon persists in finite-dimensional Banach sequence spaces, for vector-valued holomorphic mappings, and for refined Bohr–Rogosinski functionals with square-sum or energy terms (Kayumov et al., 2017, Kumar et al., 2024, Ahamed et al., 2024, Ahammed et al., 26 Aug 2025).

1. Classical baseline and the meaning of lacunarity

The classical Bohr inequality is the reference point for all subsequent developments. For $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$, the majorant series satisfies

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

and the constant $\{qs+m:s\ge 1\}$0 is sharp. In the lacunary setting, one restricts the support of the expansion to a sparse family of degrees. In the recent Banach-space literature, the dominant model is arithmetic-progression lacunarity,

$\{qs+m:s\ge 1\}$1

or equivalently $\{qs+m:s\ge 1\}$2, with $\{qs+m:s\ge 1\}$3 and $\{qs+m:s\ge 1\}$4. This includes odd, even, and fixed congruence-class degrees as special cases (Kumar et al., 2024, Ahammed et al., 26 Aug 2025).

This notion is distinct from general Hadamard-gap lacunarity. Several of the cited papers explicitly emphasize that the results are proved for arithmetic-progression support rather than for arbitrary sparse sets. A recurring theme is that removing intermediate degrees reduces the cumulative size of the majorant, so the Bohr radius can exceed the classical value $\{qs+m:s\ge 1\}$5. In one variable this enlargement can be dramatic: for bounded analytic odd functions, corresponding to $\{qs+m:s\ge 1\}$6, the sharp Bohr radius becomes $\{qs+m:s\ge 1\}$7 (Kayumov et al., 2017).

2. Banach-space and vector-valued framework

The modern formulations are expressed in terms of Fréchet expansions on Banach balls. A standard setting is the finite-dimensional complex Banach sequence space $\{qs+m:s\ge 1\}$8, $\{qs+m:s\ge 1\}$9, with norm

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

and open unit ball

$f(z)=\sum_{n\ge 0}a_n z^n$1

A holomorphic mapping $f(z)=\sum_{n\ge 0}a_n z^n$2 has the Fréchet expansion

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

where $f(z)=\sum_{n\ge 0}a_n z^n$4 is the $f(z)=\sum_{n\ge 0}a_n z^n$5-th Fréchet derivative, viewed as a continuous symmetric $f(z)=\sum_{n\ge 0}a_n z^n$6-linear map (Ahammed et al., 26 Aug 2025, Ahamed et al., 2024).

For vector-valued Bohr inequalities, a common target is the closed unit polydisk $f(z)=\sum_{n\ge 0}a_n z^n$7 endowed with the sup-norm

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

Lacunarity is then imposed degreewise on the homogeneous pieces in the Fréchet expansion. In this formulation, a vector-valued lacunary series has the form

$f(z)=\sum_{n\ge 0}a_n z^n$9

or, equivalently,

$\mathbb D$0

The geometric input is usually a Schwarz mapping $\mathbb D$1 with a zero of order $\mathbb D$2 at the origin, for which the Banach-space Schwarz lemma yields

$\mathbb D$3

That estimate is fundamental in mixed Bohr and Bohr–Rogosinski inequalities because it converts the order of vanishing of $\mathbb D$4 into an effective reduction of the radial parameter (Ahammed et al., 26 Aug 2025).

3. Sharp radii for arithmetic-progression lacunary series

The best-known scalar one-variable theorem in this direction concerns bounded analytic functions of the form

$\mathbb D$5

The sharp radius $\mathbb D$6 is the largest positive solution of

$\mathbb D$7

and

$\mathbb D$8

For $\mathbb D$9, the sharp radius is $\|f\|_\infty\le 1$0; if in addition $\|f\|_\infty\le 1$1, the radius increases to $\|f\|_\infty\le 1$2 (Kayumov et al., 2017).

In higher-dimensional sequence spaces, exact sharp radii also arise for arithmetic lacunarity. For scalar- and vector-valued holomorphic mappings on $\|f\|_\infty\le 1$3, under the structural hypotheses used in the paper, the sharp radius is the unique root in $\|f\|_\infty\le 1$4 of

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

The same equation governs both the scalar-target and vector-valued lacunary inequalities in that setting, and the radii are dimension-free in the sense that they do not explicitly depend on $\|f\|_\infty\le 1$6 (Kumar et al., 2024).

A later refined Banach-space formulation replaces the basic majorant by a sum with squared-coefficient corrections. In that setting the sharp lacunary radius $\|f\|_\infty\le 1$7 is the maximal positive root of

$\|f\|_\infty\le 1$8

For $\|f\|_\infty\le 1$9, one has $\sum_{n\ge 0}|a_n|r^n\le 1$0 for every $\sum_{n\ge 0}|a_n|r^n\le 1$1 (Ahamed et al., 2024).

A representative comparison of sharp radii is given below.

Setting	Sharp radius or defining equation	Source
Analytic $\sum_{n\ge 0}|a_n|r^n\le 1$2, $\sum_{n\ge 0}|a_n|r^n\le 1$3	$\sum_{n\ge 0}|a_n|r^n\le 1$4	(Kayumov et al., 2017)
Lacunary scalar/vector on $\sum_{n\ge 0}|a_n|r^n\le 1$5	$\sum_{n\ge 0}|a_n|r^n\le 1$6	(Kumar et al., 2024)
Refined lacunary Bohr on $\sum_{n\ge 0}|a_n|r^n\le 1$7	$\sum_{n\ge 0}|a_n|r^n\le 1$8	(Ahamed et al., 2024)
Mixed vector-valued corollary	$\sum_{n\ge 0}|a_n|r^n\le 1$9	(Ahammed et al., 26 Aug 2025)

These formulas make explicit the quantitative effect of sparsity. As the gap parameter increases, the radius equations involve higher powers of $0\le r\le 1/3$0, and the admissible radius typically grows. This suggests that arithmetic lacunarity acts as a structural damping mechanism on the majorant series.

4. Vector-valued, mixed, and refined Bohr inequalities

The 2025 vector-valued treatment on $0\le r\le 1/3$1 studies holomorphic mappings

$0\le r\le 1/3$2

with $0\le r\le 1/3$3 and $0\le r\le 1/3$4. Its principal mixed theorem combines three ingredients: a point evaluation term $0\le r\le 1/3$5, a lacunary majorant over degrees $0\le r\le 1/3$6, and a shift term $0\le r\le 1/3$7, where $0\le r\le 1/3$8 and $0\le r\le 1/3$9 are Schwarz mappings of prescribed orders. The resulting radius is sharp and is characterized as the minimal root of an explicit algebraic equation in $1/3$0 involving the parameters $1/3$1. A corollary gives

$1/3$2

and $1/3$3 is sharp (Ahammed et al., 26 Aug 2025).

Refined versions add square-sum corrections. One such functional is

$1/3$4

Together with a Schwarz-shift term, this yields a sharp refined Bohr inequality up to a radius $1/3$5 determined by an explicit root equation (Ahammed et al., 26 Aug 2025).

A related Banach-space theory for scalar-valued mappings $1/3$6 and vector-valued mappings $1/3$7 introduces refined sums such as

$1/3$8

with sharp constant $1/3$9, and analogous functional-type and norm-type lacunary versions governed by the radius $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$0 above (Ahamed et al., 2024). Earlier multidimensional work on $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$1 and on balanced Banach domains established sharp refined norm-type inequalities under restricted coordinate hypotheses, including the sharp constant $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$2 for general lacunary expansions without a constant term (Ahammed et al., 2023).

5. Bohr–Rogosinski, alternating variants, and energy functionals

The Bohr–Rogosinski extension controls a tail majorant together with a point evaluation. In the vector-valued Banach-sequence-space setting, the basic Rogosinski functional $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$3 starts at degree $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$4 and satisfies the sharp estimate

$$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$5

This leads to inequalities of the form

$$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$6

with a sharp radius defined as the unique root of an explicit equation in $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$7 depending on $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$8, $$f(z)=\sum_{n=0}^\infty a_n z^n\in H(\mathbb D,\mathbb D)$$9, and the order of vanishing of the Schwarz mapping $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$0 (Ahammed et al., 26 Aug 2025). An analogous scalar Banach-space theorem appears in the refined 2024 paper, again with a sharp root equation and the classical special cases $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$1 for $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$2 and $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$3 for $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$4 (Ahamed et al., 2024).

Another family of refinements augments the majorant by an energy term. In the vector-valued setting,

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

is combined with a polynomial $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$6. Under an explicit coefficient condition on $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$7, one obtains

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

for $$\sum_{n=0}^{\infty}|a_n|r^n\le 1 \qquad (0\le r\le 1/3),$$9, and the radius is sharp. In particular, for $\{qs+m:s\ge 1\}$00, $\{qs+m:s\ge 1\}$01, and $\{qs+m:s\ge 1\}$02 for $\{qs+m:s\ge 1\}$03,

$\{qs+m:s\ge 1\}$04

with sharp radius $\{qs+m:s\ge 1\}$05 (Ahammed et al., 26 Aug 2025).

Alternating lacunary variants form a parallel line of development. For $\{qs+m:s\ge 1\}$06 odd and $\{qs+m:s\ge 1\}$07, vector-valued alternating lacunary inequalities on $\{qs+m:s\ge 1\}$08 hold with the same sharp radius determined by

$\{qs+m:s\ge 1\}$09

while a mixed alternating majorant yields the sharp radius defined by

$\{qs+m:s\ge 1\}$10

Weighted one-variable alternating theories for arithmetic-progression subseries produce further sharp root formulas such as $\{qs+m:s\ge 1\}$11 (Kumar et al., 2024, Lin et al., 2021).

6. Extremals, dimension effects, limitations, and open problems

Sharpness is established throughout by explicit Möbius-type extremals. In one variable and in arithmetic-progression classes, the standard extremal is

$\{qs+m:s\ge 1\}$12

or equivalent Blaschke-type variants. These attain equality in the sharp radius calculations for scalar lacunary Bohr inequalities (Kayumov et al., 2017). In Banach spaces, extremals are built from norm-attaining functionals $\{qs+m:s\ge 1\}$13 and the same Möbius mechanism,

$\{qs+m:s\ge 1\}$14

which reduce the problem to a one-dimensional slice (Ahamed et al., 2024). For vector-valued maps into $\{qs+m:s\ge 1\}$15, an extremal family is

$\{qs+m:s\ge 1\}$16

and evaluation at $\{qs+m:s\ge 1\}$17 shows failure beyond the asserted radius as $\{qs+m:s\ge 1\}$18 (Ahammed et al., 26 Aug 2025).

A notable structural feature of many recent results is their lack of explicit dimension dependence. In several theorems on $\{qs+m:s\ge 1\}$19, the radii depend on lacunarity parameters and on the order of vanishing of auxiliary Schwarz mappings, but not directly on $\{qs+m:s\ge 1\}$20. This is attributed to estimates along a single coordinate or along the scalar radial parameter $\{qs+m:s\ge 1\}$21 rather than on full coefficient counting in dimension $\{qs+m:s\ge 1\}$22 (Kumar et al., 2024, Ahammed et al., 26 Aug 2025). By contrast, unrestricted multidimensional Bohr radii $\{qs+m:s\ge 1\}$23 for the polydisk satisfy the known asymptotic behavior

$\{qs+m:s\ge 1\}$24

which shows that the lacunary and restricted-direction setting is qualitatively different (Ahammed et al., 26 Aug 2025).

The main limitations are equally clear. The strongest sharp results are proved for finite-dimensional sequence spaces $\{qs+m:s\ge 1\}$25, for targets such as $\{qs+m:s\ge 1\}$26 with the sup-norm, and for arithmetic-progression lacunarity. Broader sparse sets of degrees, infinite-dimensional domains, and general Banach targets remain largely open. Existing papers explicitly identify extensions to infinite-dimensional sequence spaces, to Banach-valued targets beyond $\{qs+m:s\ge 1\}$27, and to more general lacunary patterns such as Hadamard gaps as nontrivial problems that would likely require additional geometric hypotheses or stronger hypercontractive inequalities (Kumar et al., 2024, Ahammed et al., 26 Aug 2025).