On multidimensional Bohr radii for Banach spaces

Subhadip Pal; Vasudevarao Allu

arxiv: 2406.19865 · v3 · submitted 2024-06-28 · 🧮 math.FA · math.CV

On multidimensional Bohr radii for Banach spaces

Vasudevarao Allu , Subhadip Pal This is my paper

Pith reviewed 2026-05-23 23:56 UTC · model grok-4.3

classification 🧮 math.FA math.CV

keywords multidimensional Bohr radiiBanach spacesbounded linear operatorsholomorphic functionsasymptotic estimatesarithmetic Bohr radiusell_q^n unit ball

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The pith

Multidimensional Bohr radii for bounded linear operators between Banach spaces admit exact asymptotic estimates in both finite and infinite dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a generalized multidimensional Bohr radius for holomorphic functions on the unit ball of ell_q^n spaces, taking values in arbitrary complex Banach spaces, with special attention to bounded linear operators. It derives the precise growth rate of this radius as dimension varies, covering both the finite-dimensional and infinite-dimensional cases. The estimates are then applied to produce a concrete lower bound for the arithmetic Bohr radius. A reader would care because the radius measures the largest scaling factor under which power-series expansions remain bounded, directly affecting how far analytic continuation and operator inequalities can be pushed in several complex variables.

Core claim

The multidimensional Bohr radius for bounded linear operators between complex Banach spaces, defined via the standard supremum construction over the unit ball of ell_q^n, satisfies exact asymptotic estimates that hold uniformly for finite-dimensional targets as well as for arbitrary infinite-dimensional Banach spaces; these estimates in turn supply an explicit lower bound on the arithmetic Bohr radius.

What carries the argument

The multidimensional Bohr radius, constructed as the largest r such that the operator norm of the scaled homogeneous polynomials remains controlled by the supremum norm on the unit ball of ell_q^n.

If this is right

The asymptotic formula supplies matching upper and lower bounds that become sharp as dimension tends to infinity.
The same estimates apply verbatim when the target space is any complex Banach space, finite- or infinite-dimensional.
A positive lower bound follows immediately for the arithmetic Bohr radius.
The construction extends the classical one-variable Bohr radius to the multivariable operator setting without loss of the sharp growth rate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The uniform control in infinite dimensions suggests the radius may stabilize under passage to ultrapowers or other limit constructions in Banach space theory.
The lower bound on the arithmetic radius could be tested numerically by evaluating the relevant suprema on low-dimensional ell_q^n spaces for small q.
The same asymptotic machinery might adapt to other radii that replace the ell_q ball by different convex bodies.
If the estimates remain valid for non-linear holomorphic maps, they would give a direct comparison between linear and non-linear versions of the multidimensional radius.

Load-bearing premise

The radii are defined by the usual supremum over the unit ball of ell_q^n and the maps satisfy the standard conditions of holomorphy and boundedness on that ball.

What would settle it

An explicit computation of the multidimensional Bohr radius for the identity operator on a concrete finite-dimensional ell_q^n space whose value deviates from the claimed asymptotic formula by more than a constant factor.

read the original abstract

In this paper, we study a more general version of multidimensional Bohr radii for the holomorphic functions defined on unit ball of $\ell^n_q\,\,(1\leq q\leq \infty)$ spaces with values in arbitrary complex Banach spaces. More precisely, we study the multidimensional Bohr radii for bounded linear operators between complex Banach spaces, primarily motivated by the work of A. Defant, M. Maestre, and U. Schwarting [Adv. Math. 231 (2012), pp. 2837--2857]. We obtain the exact asymptotic estimates of multidimensional Bohr radius for both finite and infinite dimensional Banach spaces. As an application, we find the lower bound of arithmetic Bohr radius.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims exact asymptotics for multidimensional Bohr radii of operator-valued holomorphic functions on ell_q^n balls in finite and infinite dimensions, plus a lower bound for the arithmetic Bohr radius, as a direct extension of Defant-Maestre-Schwarting.

read the letter

The main takeaway is that the authors extend the 2012 Adv. Math. results on multidimensional Bohr radii to the setting of bounded linear operators between complex Banach spaces, working on the unit balls of ell_q^n for 1 ≤ q ≤ ∞. They state exact asymptotic estimates that cover both finite- and infinite-dimensional cases and derive a lower bound for the arithmetic Bohr radius as an application. This stays within the standard supremum-over-the-ball definition and the usual holomorphy and boundedness assumptions from the earlier work. The extension looks consistent with the cited framework and produces the claimed asymptotics without introducing new entities or free parameters. The motivation section aligns with the Defant-Maestre-Schwarting program, and the application to the arithmetic case follows naturally from the main estimates. The citation pattern is appropriate and does not loop back on itself. The soft spot is that the derivations are not visible here, so it is not possible to confirm whether the exactness holds without any dimension-dependent adjustments or special choices in the operator norm. The lower bound for the arithmetic radius is presented only as an application, so its sharpness relative to existing bounds is unclear from the summary. This work is aimed at the small group of researchers already studying Bohr radii for holomorphic mappings in Banach spaces. Readers outside that niche will see little of broader interest. It deserves peer review because the claims are specific, the extension is legitimate, and a referee can verify the calculations directly.

Referee Report

0 major / 2 minor

Summary. The manuscript extends the multidimensional Bohr radius framework of Defant-Maestre-Schwarting to bounded linear operators between arbitrary complex Banach spaces, with the radii defined via the standard supremum construction over the unit ball of ℓ_q^n (1≤q≤∞). It claims to derive exact asymptotic estimates of these radii in both finite- and infinite-dimensional settings and, as an application, obtains a lower bound on the arithmetic Bohr radius.

Significance. If the claimed exact asymptotics hold, the work supplies precise characterizations in the operator-valued holomorphic setting that strengthen the Defant-Maestre-Schwarting theory and furnish a concrete lower bound for the arithmetic variant; such parameter-free or asymptotically sharp results are valuable for further developments in infinite-dimensional holomorphy.

minor comments (2)

The abstract states that exact asymptotic estimates are obtained but does not display the leading-order expressions; adding a one-line indication of the form of the asymptotics (e.g., the dependence on n and q) would improve readability for readers scanning the front matter.
Notation for the multidimensional Bohr radius (presumably denoted something like r_{q,n}(X,Y) or similar) should be introduced explicitly in the first section with a displayed definition, even if it follows the standard supremum construction, to avoid any ambiguity when the estimates are stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for recommending acceptance of the manuscript. The report accurately summarizes the main contributions.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions

full rationale

The abstract and available context describe a direct extension of the Defant-Maestre-Schwarting framework to operator-valued holomorphic functions on ell_q^n balls, claiming exact asymptotics for finite- and infinite-dimensional cases plus a lower bound for the arithmetic Bohr radius. These claims rest on standard supremum constructions over unit balls and usual holomorphy/boundedness conditions, with no equations, fitted parameters, self-citations, or ansatzes presented that reduce any prediction or result to its own inputs by construction. The derivation chain is therefore self-contained against the stated definitions and external motivation, with no load-bearing step available for inspection that exhibits circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.0 · 5637 in / 961 out tokens · 25009 ms · 2026-05-23T23:56:27.697007+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

[1]

Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc

L. Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc. Amer. Math. Soc. 128 (2000), 1147–1155

work page 2000
[2]

Aizenberg, A

L. Aizenberg, A. Aytuna , and P. Djakov , An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (2000), 2611–2619

work page 2000
[3]

Aizenberg, A

L. Aizenberg, A. Aytuna and P. Djakov , Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), 429–447

work page 2001
[4]

V. Allu, H. Halder, and S. Pal , Multidimensional Bohr radii for vector-valued holomorph ic functions, see https://arxiv.org/abs/2308.07825 (2023)

work page internal anchor Pith review Pith/arXiv arXiv 2023
[5]

Bayart, D

F. Bayart, D. Pellegrino , and J. B. Seoane-Sep ´Ul veda, The Bohr radius of the n-dimensional polydisk is equivalent to √ (log n)/n, Adv. Math. 264 (2014), 726–746

work page 2014
[6]

Bayart, A

F. Bayart, A. Def ant , and S. Schlüters , Monomial convergence for holomorphic functions on ℓr, J. Anal. Math. 138 (2019), 107–134

work page 2019
[7]

Bénéteau , A

C. Bénéteau , A. Dahlner and D. Kha vinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4(1) (2004), 1–19

work page 2004
[8]

Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper

O. Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper. Theory Adv. Appl., 201, Birkhäuser Verlag, Basel, 20 10

work page
[9]

Blasco , The p-Bohr radius of a Banach space, Collect

O. Blasco , The p-Bohr radius of a Banach space, Collect. Math. 68 (2017), 87–100

work page 2017
[10]

H. P. Boas and D. Kha vinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979

work page 1997
[11]

H. P. Boas , Majorant Series, J. Korean Math. Soc. 37 (2000), 321–337

work page 2000
[12]

Bohr , A theorem concerning power series, Proc

H. Bohr , A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1–5

work page 1914
[13]

Bombal, D

F. Bombal, D. Pérez-García and I. Villanuev a, Multilinear extensions of Grothendieck’s theo- rem, Q. J. Math 55 (2004), 441–450

work page 2004
[14]

Bombieri, Sopra un teorema di H

E. Bombieri, Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggior anti delle serie di potenze, Boll. Un. Mat. Ital. 17 (1962), 276–282

work page 1962
[15]

Bombieri and J

E. Bombieri and J. Bourgain , A remark on Bohr’s inequality, Internat. Math. Res. Notices 80 (2004), 4307–4330

work page 2004
[16]

Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad

N. Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad. Math. Bull. 66 (2023), 682–699

work page 2023
[17]

Das , The p-Bohr radius for vector-valued holomorphic and pluriharmo nic functions, Forum Math

N. Das , The p-Bohr radius for vector-valued holomorphic and pluriharmo nic functions, Forum Math. 36 (2024), 765–782

work page 2024
[18]

Def ant and L

A. Def ant and L. Frerick , A logarithmic lower bound for multi-dimenional Bohr radii , Israel J. Math. 152 (2006), 17–28

work page 2006
[19]

Def ant and L

A. Def ant and L. Frerick , The Bohr radius of the unit ball of ℓn p , J. Reine Angew. Math. 660 (2011), 131–147. On multidimensional Bohr radii for Banach spaces 19

work page 2011
[20]

Def ant, L

A. Def ant, L. Frerick, J. Ortega-Cerd `A, M. Ounaïes , and K. Seip , The Bohnenblust-Hille inequality for homogeneous polynomils in hypercontractiv e, Ann. of Math. 174 (2011), 512–517

work page 2011
[21]

Def ant, D

A. Def ant, D. García , and M. Maestre , Bohr power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197

work page 2003
[22]

Def ant, D

A. Def ant, D. García, M. Maestre , and P. Sevilla-Peris , Dirichlet Series and Holomorphic Functions in High Dimensions , New Mathematical Monographs: 37, Cambridge University Press, Cambridge, (2019)

work page 2019
[23]

Def ant, M

A. Def ant, M. Maestre , and C. Prengel , The arithmetic Bohr radius, Q. J. Math. 59 (2008), 189–205

work page 2008
[24]

Def ant, M

A. Def ant, M. Maestre , and C. Prengel , Domains of convergence for monomial expansions of holomorphic functions in inﬁnitely many variables, J. Reine Angew. Math. 634 (2009), 13–49

work page 2009
[25]

Def ant, M

A. Def ant, M. Maestre and U. Schw arting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857

work page 2012
[26]

Diestel, H

J. Diestel, H. Jarchow , and A. Tonge , Absolutely Summing Operators , in: Cambridge Studies in Advance Mathematics 43, Cambridge University Press, Cambridge, (1995)

work page 1995
[27]

P. G. Dixon , Banach algebras satisfying the non-unital von Neumann ine quality, Bull. Lond. Math. Soc. 27 (4) (1995), 359–362

work page 1995
[28]

P. B. Djakov and M. S. Ramanujan , A remark on Bohr’s theorem and its generalizations, J. Anal. 8 (2000), 65–77

work page 2000
[29]

Dineen and R

S. Dineen and R. M. Timoney , Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227–234

work page 1989
[30]

Galicer, M

D. Galicer, M. Mansilla and S. Muro , Mixed Bohr radius in several variables, Trans. Amer. Math. Soc. 373 (2020), 777–796

work page 2020
[31]

Hamada, T

H. Hamada, T. Honda and G. Kohr , Bohr’s theorem for holomorphic mappings with values in homogeneous balls, Israel J. Math. 173 (2009), 177–187

work page 2009
[32]

Kumar and R

S. Kumar and R. Manna , Multi-dimensional Bohr radii of Banach space valued holom orphic func- tions, see https://arxiv.org/pdf/2303.17416 (2023)

work page arXiv 2023
[33]

Lindenstrauss and L

J. Lindenstrauss and L. Tzafriri , Classical Banach Spaces I: Sequence Spaces , in: Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Springer-Verlag, Berlin-Heidelberg-New York, (1977)

work page 1977
[34]

Lindenstrauss and L

J. Lindenstrauss and L. Tzafriri , Classical Banach Spaces II: Function Spaces , in: Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer-Verlag, Berlin-Heidelberg-New York, (1979)

work page 1979
[35]

V. I. Paulsen, G. Popescu and D. Singh , On Bohr’s inequality, Proc. Lond. Math. Soc. s3-85 (2002), 493–512

work page 2002
[36]

Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv

G. Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv. Math. 347 (2019), 1002–1053

work page 2019
[37]

Sidon , Uber einen satz von Hernn Bohr, Math

S. Sidon , Uber einen satz von Hernn Bohr, Math. Zeit. 26 (1927), 731–732

work page 1927
[38]

Tomic , Sur un theoreme de H

M. Tomic , Sur un theoreme de H. Bohr, Math. Scand. 11 (1962), 103–106. V asudev arao Allu, School of Basic Sciences, Indian Institute of Technology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. Email address : avrao@iitbbs.ac.in Subhadip Pal, School of Basic Sciences, Indian Institute of Tec hnology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. E...

work page 1962

[1] [1]

Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc

L. Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc. Amer. Math. Soc. 128 (2000), 1147–1155

work page 2000

[2] [2]

Aizenberg, A

L. Aizenberg, A. Aytuna , and P. Djakov , An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (2000), 2611–2619

work page 2000

[3] [3]

Aizenberg, A

L. Aizenberg, A. Aytuna and P. Djakov , Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), 429–447

work page 2001

[4] [4]

V. Allu, H. Halder, and S. Pal , Multidimensional Bohr radii for vector-valued holomorph ic functions, see https://arxiv.org/abs/2308.07825 (2023)

work page internal anchor Pith review Pith/arXiv arXiv 2023

[5] [5]

Bayart, D

F. Bayart, D. Pellegrino , and J. B. Seoane-Sep ´Ul veda, The Bohr radius of the n-dimensional polydisk is equivalent to √ (log n)/n, Adv. Math. 264 (2014), 726–746

work page 2014

[6] [6]

Bayart, A

F. Bayart, A. Def ant , and S. Schlüters , Monomial convergence for holomorphic functions on ℓr, J. Anal. Math. 138 (2019), 107–134

work page 2019

[7] [7]

Bénéteau , A

C. Bénéteau , A. Dahlner and D. Kha vinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4(1) (2004), 1–19

work page 2004

[8] [8]

Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper

O. Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper. Theory Adv. Appl., 201, Birkhäuser Verlag, Basel, 20 10

work page

[9] [9]

Blasco , The p-Bohr radius of a Banach space, Collect

O. Blasco , The p-Bohr radius of a Banach space, Collect. Math. 68 (2017), 87–100

work page 2017

[10] [10]

H. P. Boas and D. Kha vinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979

work page 1997

[11] [11]

H. P. Boas , Majorant Series, J. Korean Math. Soc. 37 (2000), 321–337

work page 2000

[12] [12]

Bohr , A theorem concerning power series, Proc

H. Bohr , A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1–5

work page 1914

[13] [13]

Bombal, D

F. Bombal, D. Pérez-García and I. Villanuev a, Multilinear extensions of Grothendieck’s theo- rem, Q. J. Math 55 (2004), 441–450

work page 2004

[14] [14]

Bombieri, Sopra un teorema di H

E. Bombieri, Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggior anti delle serie di potenze, Boll. Un. Mat. Ital. 17 (1962), 276–282

work page 1962

[15] [15]

Bombieri and J

E. Bombieri and J. Bourgain , A remark on Bohr’s inequality, Internat. Math. Res. Notices 80 (2004), 4307–4330

work page 2004

[16] [16]

Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad

N. Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad. Math. Bull. 66 (2023), 682–699

work page 2023

[17] [17]

Das , The p-Bohr radius for vector-valued holomorphic and pluriharmo nic functions, Forum Math

N. Das , The p-Bohr radius for vector-valued holomorphic and pluriharmo nic functions, Forum Math. 36 (2024), 765–782

work page 2024

[18] [18]

Def ant and L

A. Def ant and L. Frerick , A logarithmic lower bound for multi-dimenional Bohr radii , Israel J. Math. 152 (2006), 17–28

work page 2006

[19] [19]

Def ant and L

A. Def ant and L. Frerick , The Bohr radius of the unit ball of ℓn p , J. Reine Angew. Math. 660 (2011), 131–147. On multidimensional Bohr radii for Banach spaces 19

work page 2011

[20] [20]

Def ant, L

A. Def ant, L. Frerick, J. Ortega-Cerd `A, M. Ounaïes , and K. Seip , The Bohnenblust-Hille inequality for homogeneous polynomils in hypercontractiv e, Ann. of Math. 174 (2011), 512–517

work page 2011

[21] [21]

Def ant, D

A. Def ant, D. García , and M. Maestre , Bohr power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197

work page 2003

[22] [22]

Def ant, D

A. Def ant, D. García, M. Maestre , and P. Sevilla-Peris , Dirichlet Series and Holomorphic Functions in High Dimensions , New Mathematical Monographs: 37, Cambridge University Press, Cambridge, (2019)

work page 2019

[23] [23]

Def ant, M

A. Def ant, M. Maestre , and C. Prengel , The arithmetic Bohr radius, Q. J. Math. 59 (2008), 189–205

work page 2008

[24] [24]

Def ant, M

A. Def ant, M. Maestre , and C. Prengel , Domains of convergence for monomial expansions of holomorphic functions in inﬁnitely many variables, J. Reine Angew. Math. 634 (2009), 13–49

work page 2009

[25] [25]

Def ant, M

A. Def ant, M. Maestre and U. Schw arting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857

work page 2012

[26] [26]

Diestel, H

J. Diestel, H. Jarchow , and A. Tonge , Absolutely Summing Operators , in: Cambridge Studies in Advance Mathematics 43, Cambridge University Press, Cambridge, (1995)

work page 1995

[27] [27]

P. G. Dixon , Banach algebras satisfying the non-unital von Neumann ine quality, Bull. Lond. Math. Soc. 27 (4) (1995), 359–362

work page 1995

[28] [28]

P. B. Djakov and M. S. Ramanujan , A remark on Bohr’s theorem and its generalizations, J. Anal. 8 (2000), 65–77

work page 2000

[29] [29]

Dineen and R

S. Dineen and R. M. Timoney , Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227–234

work page 1989

[30] [30]

Galicer, M

D. Galicer, M. Mansilla and S. Muro , Mixed Bohr radius in several variables, Trans. Amer. Math. Soc. 373 (2020), 777–796

work page 2020

[31] [31]

Hamada, T

H. Hamada, T. Honda and G. Kohr , Bohr’s theorem for holomorphic mappings with values in homogeneous balls, Israel J. Math. 173 (2009), 177–187

work page 2009

[32] [32]

Kumar and R

S. Kumar and R. Manna , Multi-dimensional Bohr radii of Banach space valued holom orphic func- tions, see https://arxiv.org/pdf/2303.17416 (2023)

work page arXiv 2023

[33] [33]

Lindenstrauss and L

J. Lindenstrauss and L. Tzafriri , Classical Banach Spaces I: Sequence Spaces , in: Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Springer-Verlag, Berlin-Heidelberg-New York, (1977)

work page 1977

[34] [34]

Lindenstrauss and L

J. Lindenstrauss and L. Tzafriri , Classical Banach Spaces II: Function Spaces , in: Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer-Verlag, Berlin-Heidelberg-New York, (1979)

work page 1979

[35] [35]

V. I. Paulsen, G. Popescu and D. Singh , On Bohr’s inequality, Proc. Lond. Math. Soc. s3-85 (2002), 493–512

work page 2002

[36] [36]

Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv

G. Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv. Math. 347 (2019), 1002–1053

work page 2019

[37] [37]

Sidon , Uber einen satz von Hernn Bohr, Math

S. Sidon , Uber einen satz von Hernn Bohr, Math. Zeit. 26 (1927), 731–732

work page 1927

[38] [38]

Tomic , Sur un theoreme de H

M. Tomic , Sur un theoreme de H. Bohr, Math. Scand. 11 (1962), 103–106. V asudev arao Allu, School of Basic Sciences, Indian Institute of Technology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. Email address : avrao@iitbbs.ac.in Subhadip Pal, School of Basic Sciences, Indian Institute of Tec hnology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. E...

work page 1962