Advertisement

Mathematical Notes

, Volume 66, Issue 2, pp 223–232 | Cite as

On certain relations between the maximum modulus and the maximal term of an entire dirichlet series

  • O. B. Skaskiv
Article

Abstract

For an entire Dirichlet series\(F(z){\text{ = }}\sum\nolimits_{{\text{n = 0}}}^{{\text{ + }}\infty } {ane^{{\text{z}}\lambda {\text{n}}} ,{\text{ }}0 \leqslant \lambda n \uparrow + \infty {\text{ }}(n \to + \infty ),} \), sufficient conditions on the exponents\(\lambda _n \) are established such that the following relations hold outside a set of finite measure asx→+∞:
$$\psi (In sup\left\{ {|F(x + iy)|:y \in \mathbb{R}} \right\}) = (1 + o(1))\psi (In max\{ |a_n |e^{x\lambda n} :n \geqslant \} ),$$
, where ψ(x) is a function increasing to +∞ and such thatx≤ψ(x)≤e x (x≥0).

Key words

entire Dirichlet series maximum modulus of a Dirichlet series maximal term of a Dirichlet series set of finite measure Chebyshev’s inequality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. B. Skaskiv, “The maximum modulus and the maximal term of the entire Dirichlet series,”Dokl. Akad. Nauk Ukrain. SSR Ser. A, No. 11, 22–24 (1984).Google Scholar
  2. 2.
    O. B. Skaskiv, “On the behavior of the maximal term of the Dirichlet series defining an entire function,”Mat. Zametki [Math. Notes],37, No. 1, 41–47 (1985).zbMATHGoogle Scholar
  3. 3.
    M. M. Khomyak, “On the maximal term of the Dirichlet series defining an entire function,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], No. 10, 79–81 (1982).Google Scholar
  4. 4.
    M. N. Sheremeta, “On the behavior of the maximum the modulus of an entire Dirichlet series outside an exceptional set,”Mat. Zametki [Math. Notes],57, No. 2, 283–296 (1995).Google Scholar
  5. 5.
    W. K. Hayman,Subharmonic Functions, Vol. 2, London Math. Soc. Monographs, Vol. 20, Acad. Press, London (1989).zbMATHGoogle Scholar
  6. 6.
    Open Problems, Matem. studii. Pratsi Lviv. matem. tovarishch. No. 3, Lvov (1994).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • O. B. Skaskiv
    • 1
  1. 1.I. Franko Lvov State UniversityUSSR

Personalised recommendations