# On the maximum modulus of entire functions

On the maximum modulus of entire functions By P. ERD()S (Budapest), corresponding member of the Academy, and T. KOv/~RI (Budapest) The function 34(i") = max If(z)[ I~--r is called the maximum modulus function of the entire function f(z). In the present paper we discuss the approximation of maximum modulus functions, by means of power series with positive coefficients. We shall prove the fol- lowing THEOREM I. For every M(r) there exists a power series N(r)~-Xc.r '~ ]vith non-negative coefficients and with the property l M(r) 6< U(~ < 3. Though these constants are not the best possible, the theorem can not be sharpened essentially. We shall show this by constructing a maximum modulus function M(r) with the property that there does not exist a power series N(f) with non-negative coefficients which would satisfy the following asymptotic equality : .44 (f) ,-~ N(r). in fact, the following stronger result holds: THEOREM II. There exists an absohtte constant ~,~ > 0 ~o ~-~j and a ( 'i maximum modulus function M(f) so that for every power series N(r) with non-negative coefficients the inequality M(r) fails for arbitrary false 1". It is to be hoped that by the aid of Theorem I it will be possible to extend certain http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

# On the maximum modulus of entire functions

, Volume 7 (4) – Jul 16, 2005
13 pages      /lp/springer-journals/on-the-maximum-modulus-of-entire-functions-z70kNjdp9y
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF02020527
Publisher site
See Article on Publisher Site

### Abstract

By P. ERD()S (Budapest), corresponding member of the Academy, and T. KOv/~RI (Budapest) The function 34(i") = max If(z)[ I~--r is called the maximum modulus function of the entire function f(z). In the present paper we discuss the approximation of maximum modulus functions, by means of power series with positive coefficients. We shall prove the fol- lowing THEOREM I. For every M(r) there exists a power series N(r)~-Xc.r '~ ]vith non-negative coefficients and with the property l M(r) 6< U(~ < 3. Though these constants are not the best possible, the theorem can not be sharpened essentially. We shall show this by constructing a maximum modulus function M(r) with the property that there does not exist a power series N(f) with non-negative coefficients which would satisfy the following asymptotic equality : .44 (f) ,-~ N(r). in fact, the following stronger result holds: THEOREM II. There exists an absohtte constant ~,~ > 0 ~o ~-~j and a ( 'i maximum modulus function M(f) so that for every power series N(r) with non-negative coefficients the inequality M(r) fails for arbitrary false 1". It is to be hoped that by the aid of Theorem I it will be possible to extend certain

### Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Jul 16, 2005