A note on the Wiman–Valiron inequality

[en] The Wiman–Valiron inequality relates the maximum modulus of an analytic function to its Taylor coefficients via the maximum term. After a short overview of the known results, we obtain a general version of this inequality that seems to have been overlooked in the literature so far. We end the paper with an open problem.

Research center :

CREMMI - Modélisation mathématique et informatique

Disciplines :

Mathematics

Author, co-author :

Grosse-Erdmann, Karl ; Université de Mons - UMONS > Faculté des Sciences > Service d'Analyse fonctionnelle

Language :

English

Title :

A note on the Wiman–Valiron inequality

Publication date :

01 November 2024

Journal title :

Archiv der Mathematik

ISSN :

0003-889X

eISSN :

1420-8938

Publisher :

Birkhauser

Volume :

124

Issue :

Pages :

63-74

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://link.springer.com/content/pdf/10.1007/s00013-024-02061-2.pdf

Research unit :

S844 - Probabilité et statistique

Research institute :

Complexys

Funders :

Fonds De La Recherche Scientifique - FNRS

Funding number :

CDR J.0078.21

Funding text :

The author is supported by the Fonds de la Recherche Scientifique\u2014FNRS under Grant no CDR J.0078.21.

Available on ORBi UMONS :

since 13 January 2025

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Bibliography

Agneessens, K., Grosse-Erdmann, K.-G.: Rate of growth of random analytic functions, with an application to linear dynamics. arXiv:2409.04235 (2024)
P.C. Fenton M.M. Strumia Wiman-Valiron theory in the disc J. Lond. Math. Soc. (2) 2009 79 478 496 2496525 10.1112/jlms/jdp001
W.K. Hayman The local growth of power series: a survey of the Wiman-Valiron method Canad. Math. Bull. 1974 17 317 358 385095 10.4153/CMB-1974-064-0
Hayman, W.K.: Subharmonic Functions. Vol. 2. Academic Press Inc., London (1989)
T. Kovari On the maximum modulus and maximum term of functions analytic in the unit disc J. London Math. Soc. 1966 41 129 137 185090 10.1112/jlms/s1-41.1.129
A.O. Kuryliak O.B. Skaskiv Sub-Gaussian random variables and Wiman’s inequality for analytic functions Carpathian Math. Publ. 2023 15 306 314 4611939 10.15330/cmp.15.1.306-314
Rosenbloom, P.C.: Probability and entire functions. In: Studies in Mathematical Analysis and Related Topics, pp. 325–332. Stanford Univ. Press, Stanford, CA (1962)
Rudin, W.: Real and Complex Analysis. Third Edition. McGraw-Hill Book Co., New York (1987)
O.B. Skaskiv A.O. Kuryliak Wiman’s type inequality for analytic and entire functions and h-measure of an exceptional sets Carpathian Math. Publ. 2020 12 492 498 4223779 10.15330/cmp.12.2.492-498
Suleĭmanov, N.M.: Wiman-Valiron-type estimates for power series with a finite radius of convergence and their exactness (Russian). Dokl. Akad. Nauk SSSR 253, 822–824 (1980). English translation in: Soviet Math. Dokl. 22, 190–192 (1980)
G. Valiron Sur le maximum du module des fonctions entiéres C. R. Acad. Sci. Paris 1918 166 605 608
Valiron, G.: Les théorèmes généraux de M. Borel dans la théorie des fonctions entières. Ann Sci. Ecole Norm. Sup. (3) 37, 219–253 (1920)
Valiron, G.: Lectures on the General Theory of Integral Functions. Deighton, Bell and Co., Cambridge (1923)
A. Wiman Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Gliede der zugehörigen Taylor’schen Reihe Acta Math. 1914 37 305 326 1555102 10.1007/BF02401837