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Bohr radius for Banach spaces on simply connected domains

Part of: General theory of linear operators Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices Series expansions

Published online by Cambridge University Press:  03 November 2023

Vasudevarao Allu Open the ORCID record for Vasudevarao Allu [Opens in a new window]  and
Himadri Halder
Vasudevarao Allu
Affiliation:
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar, Odisha, India (avrao@iitbbs.ac.in; himadrihalder119@gmail.com)
Himadri Halder
Affiliation:
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar, Odisha, India (avrao@iitbbs.ac.in; himadrihalder119@gmail.com)
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Abstract

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q \lt \infty$, we denote

\begin{equation*}R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \left\lVert x_{0}\right\rVert^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \left\lVert x_{n}\right\rVert\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\}\end{equation*}
and define the Bohr radius associated with ϕ by
\begin{equation*}R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1\right\}.\end{equation*}
In this article, we extensively study the Bohr radius $R_{p,q,\phi}(\Omega,X)$, when X is an arbitrary Banach space, and $X=\mathcal{B}(\mathcal{H})$ is the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator.

Keywords

Banach spaceoperator valuedsimply connected domainsBohr radiusCesáro operatorBernardi operator

MSC classification

Primary: 46E40: Spaces of vector- and operator-valued functions 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 47A63: Operator inequalities
Secondary: 46B20: Geometry and structure of normed linear spaces 30B10: Power series (including lacunary series)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 113 - 141
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Ahamed, M. B., Allu, V. and Halder, H., The Bohr phenomenon for analytic functions on simply connected domains, Ann. Acad. Sci. Fenn. Math. (2021), To appear.CrossRefGoogle Scholar
Aizenberg, L., Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), 1147–1155.CrossRefGoogle Scholar
Aizenberg, L., Generalization of results about the Bohr radius for power series, Stud. Math. 180 (2007), 161–168.CrossRefGoogle Scholar
Aizenberg, L., Aytuna, A. and Djakov, P., Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), 429–447.CrossRefGoogle Scholar
Albanese, A. A., Bonet, J. and Ricker, J. W., The Cesáro operator on power series spaces, Stud. Math. 240 (2018), 47–68.CrossRefGoogle Scholar
Alkhaleefah, S. A., Kayumov, I. R. and Ponnusamy, S., On the Bohr inequality with a fixed zero coefficient, Proc. Amer. Math. Soc. 147 (2019), 5263–5274.CrossRefGoogle Scholar
Allu, V. and Halder, H., Bhor phenomenon for certain subclasses of Harmonic Mappings, Bull. Sci. Math. 173 (2021), .CrossRefGoogle Scholar
Allu, V. and Halder, H., Bohr radius for certain classes of starlike and convex univalent functions, J. Math. Anal. Appl. 493(1) (2021), .CrossRefGoogle Scholar
Anderson, J. M. and Rovnyak, J., On generalized Schwarz-Pick estimates, Mathematika 53 (2006), 161–168.CrossRefGoogle Scholar
Aytuna, A. and Djakov, P., Bohr property of bases in the space of entire functions and its generalizations, Bull. Lond. Math. Soc. 45(2) (2013), 411–420.CrossRefGoogle Scholar
Bénéteau, C., Dahlner, A. and Khavinson, D., Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4(1) (2004), 1–19.CrossRefGoogle Scholar
Bayart, F., Pellegrino, D. and Seoane-Sepúlveda, J. B., The Bohr radius of the n-dimensional polydisk is equivalent to $\sqrt(log \, n)/n$, Adv. Math. 264 (2014), 726–746.CrossRefGoogle Scholar
Bhowmik, B. and Das, N., Bohr phenomenon for operator-valued functions, Proc. Edinb. Math. Soc. 10.1017/S0013091520000395.Google Scholar
Blasco, O., The Bohr radius of a Banach space, in Vector Measures, Integration and Related Topics, Operator Theory: Advances and Applications, Volume 201 (Birkhäuser Verlag, Basel, 2010).Google Scholar
Blasco, O., The p-Bohr radius of a Banach space, Collect. Math. 68 (2017), 87–100.CrossRefGoogle Scholar
Boas, H. P. and Khavinson, D., Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979.CrossRefGoogle Scholar
Bohr, H., A theorem concerning power series, Proc. Lond. Math. Soc. (1914), 1–5.CrossRefGoogle Scholar
Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M. and Seip, K., The Bohnenblust-Hille inequality for homogeneous polynomials in hypercontractive, Ann. of Math. 174 (2011), 512–517.CrossRefGoogle Scholar
Dixon, P. G., Banach algebras satisfying the non-unital von Neumann inequality, Bull. Lond. Math. Soc. 27(4) (1995), 359–362.CrossRefGoogle Scholar
Djakov, P. B. and Ramanujan, M. S., A remark on Bohr’s theorem and its generalizations, J. Anal. 8 (2000), 65–77.Google Scholar
Duren, P. L., Univalent functions, Grundlehren der mathematischen Wisseenschaften, Volume 259 (Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983).Google Scholar
Evdoridis, S., Ponnusamy, S. and Rasila, A., Improved Bohr’s inequality for locally univalent harmonic mappings, Indag. Math. (N.S.) 30(1) (2019), 201–213.CrossRefGoogle Scholar
Evdoridis, S., Ponnusamy, S. and Rasila, A., Improved Bohr’s inequality for shifted disks, Results Math. 76(14) (2021), .CrossRefGoogle Scholar
Fournier, R. and Ruscheweyh, S.. On the Bohr radius for simply connected domains, Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes, Volume 51, 2010, .CrossRefGoogle Scholar
Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals II, Math. Z. 34 (1931), 403–439.CrossRefGoogle Scholar
Kayumov, I. R., Khammatova, D. M. and Ponnusamy, S., On the Bohr inequality for the Cesáaro operator, C. R. Math. Acad. Sci. Paris 358 (2020), 615–620.CrossRefGoogle Scholar
Kayumov, I. R., Khammatova, D. M., Ponnusamy, S., The Bohr inequality for the generalized Cesáaro averaging operators, https://arxiv.org/abs/2104.01550.Google Scholar
Kayumov, I. R. and Ponnusamy, S., On a powered Bohr inequality, Ann. Acad. Sci. Fenn. Ser. A 44 (2019), 301–310.Google Scholar
Kayumov, I. R., Ponnusamy, S. and Shakirov, N., Bohr radius for locally univalent harmonic mappings, Math. Nachr. 291 (2018), 1757–1768.CrossRefGoogle Scholar
Liu, M. S. and Ponnusamy, S., Multidimensional analogues of refined Bohr’s inequality, Proc. Amer. Math. Soc. 149 (2021), 2133–2146.CrossRefGoogle Scholar
Miller, S. S. and Mocanu, P. T., Differential subordinations: Theory and applications (Marcel Dekker, New York, 2000).CrossRefGoogle Scholar
Paulsen, V.I., Popescu, G. and Singh, D., On Bohr’s inequality, Proc. Lond. Math. Soc. (2002), 493–512.CrossRefGoogle Scholar
Popescu, G., Bohr inequalities for free holomorphic functions on polyballs, Adv. Math. 347 (2019), 1002–1053.CrossRefGoogle Scholar
Stempak, K., Cesáro averaging operators, Proc. R. Soc. Edinb. Sect. A Math. 124 (1994), 121–126.CrossRefGoogle Scholar
udez, T. B., Bonilla, A., Müller, V. and Peris, A., Cesáro bounded operators on Banach spaces, J. Anal. Math. 140 (2020), 187–206.Google Scholar