On the relative growth of Dirichlet series with zero abscissa of absolute convergence

  • O. M. Mulyava Kyiv National University of Food Technologies, Kyiv, Ukraine
Keywords: Dirichlet series, relative growth, generalized order.

Abstract

Let $F$ and $G$ be analytic functions given by Dirichlet series with exponents increasing to $+\infty$ and zero abscissa of absolute convergence.
The growth of $F$ with respect to $G$ is studied through the generalized order
$$\varrho^0_{\alpha,\beta}[F]_G=\varlimsup\limits_{\sigma\uparrow 0}\dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)}$$
and the generalized lower order $$\lambda^0_{\alpha,\beta}[F]_G=\varliminf\limits_{\sigma\uparrow 0} \dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)},$$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\mathbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma)$ and $\alpha$ and $\beta$ are positive increasing to $+\infty$ functions.
Formulas are found for the finding these quantities.

References

1. Ch. Roy, On the relative order and lower order of an entire functiion// Bull. Soc. Cal. Math. Soc.2010. – V.102, No1 – P. 17–26.
2. S.K. Data, A.R. Maji, Relative order of entire functions in terms of their maximum terms// Int. Journal of Math. Analysis. – 2011. – V.5, No43. – P. 2119–2126.
3. S.K. Data, T. Biswas, Ch. Ghosh, Growth analysis of entire functions concerning generalized relative type and generalized relative weak type// Facta Univ.(NIS). Ser. Math. Inform. – 2015. – V.30, No3.– P. 295–324.
4. S.K. Data, T. Biswas, A. Hoque, Some results on the growth analysis of entire function using their maximum terms and relative L∗ –order// Journ. Math. Extension. – 2016. – V.10, No2. – P. 59–73.
5. S.K. Data, T. Biswas, P. Das, Some results on generalized relative order of meromorohic functions// Ufa Math. Journ. – 2016. – V.8, No2. – P. 92–103.
6. S.K. Data, T. Biswas, Growth analysis of entire functions of two complex variables// Sahand Communications in Math. Analysis. – 2016. – V.3, No2. – P. 13–22.
7. S.K. Data, T. Biswas, Some growth analysis of entire functions in the form of vector valued Dirichlet series on the basis on their relative Ritt L∗ –order and relative Ritt L∗ –lower order// New Trends in Math. Sci. 2017. – V.5, No2. – P. 97–103.
8. Ja.D. Pyanylo, M.M. Sheremeta, On the growth of entire fuctions given by Dirichlet series// Izv. vuzov, Mat. – 1975. – V.10. – P. 91–93. (in Russian)
9. M.M. Sheremeta, Entire Dirichlet series. – Kyiv: ISDO. – 1993. – 168 p. (in Ukrainian)
10. O.M. Mulyava, M.M. Sheremeta, Relative growth of Dirichlet series// Mat. Stud. – 2018. – V.49, No2.P. 158–164.
11. O.M. Mulyava, M.M. Sheremeta, Remarks to relative growth of entire Dirichlet series// Visnyk of Lviv Univ. Ser. mech.–math. – 2019. – V.87. – P. 73–81.
12. Yu.M. Gal’, M.M. Sheremeta, On the growth of analytic in half–plane fuctions given by Dirichlet series// Dopovidi AN URSR, ser. A. – 1978. – No12. – P. 1065–1067. (in Russian)
13. Yu.M. Gal’, On the growth of analytic functions, represented by Dirichlet series absolutely convergent in half-plane. – Drogobych. – 1980. – 30 p. – Manuscr. Dep. in VININI 16.09.1980, No 4080–80Dep. (in Russian)
14. V.S. Boychuk, On the growth of Dirichlet series absolutely convergent in half-plane// Mat. sbornik.Kiev: Nauk. dumka. – 1976. – P. 238–240. (in Russian)
Published
2021-03-06
How to Cite
Mulyava, O. M. (2021). On the relative growth of Dirichlet series with zero abscissa of absolute convergence. Matematychni Studii, 55(1), 44-50. https://doi.org/10.30970/ms.55.1.44-50
Section
Articles