IJPAM: Volume 113, No. 1 (2017)

Title

ENTIRE DIRICHLET SERIES WITH MONOTONOUS
COEFFICIENTS AND LOGARITHMIC h-MEASURE

Authors

S.I. Panchuk$^1$, T.M. Salo$^2$, O.B. Skaskiv$^3$
$^{1,3}$Department of Mechanics and Mathematics
Ivan Franko National University of L'viv
L'viv, UKRAINE
$^2$Institute of Applied Mathematics and Fundamental Sciences
National University ``Lvivska Politekhnika''
UKRAINE

Abstract

Let $F$ be an entire function represented by absolutely convergent for all $z\in\mathbb{C}$ Dirichlet series of the form $ F(z) = \sum\nolimits_{n=0}^{+\infty} a_{n}e^{z\lambda_{n}},$ where a sequence $(\lambda_n)$ such that $\lambda_n\in\mathbb{R}$ and $(\forall n\geq 0):\ 0\leq\lambda_n<\beta:=\sup\{\lambda_j:\ j\geq0\}\leq +\infty.$ In this paper we find the condition such that the relation $F(x+iy)=(1+o(1))a_{\nu(x, F)}e^{(x+iy)\lambda_{\nu(x, F)}} $ holds as $x\to +\infty$ outside some set $E$ of finite logarithmic $h$-measure (i.e. $\text{\rm h-log-meas}(E):=\int_{E\cap[1, +\infty)}h(x) d\ln x<+\infty$) uniformly in $y\in\mathbb{R}$, where $h$ is non-decrease positive continuous function on $[0,+\infty)$.

History

Received: December 13, 2016
Revised: January 29, 2017
Published: February 28, 2017

AMS Classification, Key Words

AMS Subject Classification: 30B20, 30D20
Key Words and Phrases: entire Dirichlet series, h-measure, exceptional set

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Bibliography

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O.B. Skaskiv, On the minimum of the absolutevalue of the sum for a Dirichlet series with bounded sequence ofexponents, Math. Notes 56 (5) (1994) 1177-1184, doi: https://doi.org/10.1007/BF02274666.

2
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3
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8
T.M. Salo, O.B. Skaskiv, The minimum modulus of gap power series and h-measure of exceptional sets, arXiv: 1512.05557v2[math.CV] 21 Dec 2015, 13 p.

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How to Cite?

DOI: 10.12732/ijpam.v113i1.13 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 113
Issue: 1
Pages: 141 - 149


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