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Acta Mathematica Academiae Scientiarum Hungarica
, Volume 7 (4) – Jul 16, 2005

/lp/springer-journals/on-the-maximum-modulus-of-entire-functions-z70kNjdp9y

- Publisher
- Springer Journals
- Copyright
- Copyright
- Subject
- Mathematics; Mathematics, general
- ISSN
- 0001-5954
- eISSN
- 1588-2632
- DOI
- 10.1007/BF02020527
- Publisher site
- See Article on Publisher Site

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

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

**Published: ** Jul 16, 2005

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