Relative growth of Dirichlet series with different abscissas of absolute convergence

Abstract

UDC 517.537.72

We study the growth of a Dirichlet series $F(s)=\sum _{n=1}^{\infty}f_n\exp\{s\lambda_n\}$ with zero abscissa of absolute convergence with respect to the entire Dirichlet series $G(s)=\sum _{n=1}^{\infty}g_n\exp\{s\lambda_n\}$ by using the generalized quantities of order $\varrho^0_{\beta,\beta}[F]_G=\varlimsup\nolimits_{\sigma\uparrow 0}\dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)}$ and lower order $\lambda^0_{\beta,\beta}[F]_G=\varliminf_{\sigma\uparrow 0} \dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)},$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|\colon t\in{\Bbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma),$ and $\beta$ is a positive increasing function growing to $+\infty.$

 

References

Ch. Roy, On the relative order and lower order of an entire functiion, Bull. Calcutta Math. Soc., 102, № 1, 17 – 26 (2010).

S. K. Data, A. R. Maji, Relative order of entire functions in terms of their maximum terms, Int. J. Math. and Anal., 5, № 43, 2119 – 2126 (2011).

S. K. Data, T. Biswas, Ch. Ghosh, Growth analysis of entire functions concerning generalized relative type and generalized relative weak type, Facta Univ. Ser. Math. and Inform., 30, № 3, 295 – 324 (2015).

S. K. Data, T. Biswas, A. Hoque, Some results on the growth analysis of entire function using their maximum terms and relative $L^{ast}$ -order, J. Math. Ext., 10, № 2, 59 – 73 (2016).

S. K. Data, T. Biswas, P. Das, Some results on generalized relative order of meromorohic functions, Ufa Math. J., 8,№ 2, 92 – 103 (2016), https://doi.org/10.13108/2016-8-2-95

S. K. Data, T. Biswas, Growth analysis of entire functions of two complex variables, Sahad Commun. Math. Anal.,3, Issue 2, 13 – 22 (2016).

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 $L^{ast}$-order and relative Ritt $L^{ast}$-lower order, New Trends Math. Sci., 5, № 2, 97 –103 (2017).

Ya. D. P`yany`lo, M. N. Sheremeta, O roste czely`kh funkczij, predstavlenny`kh ryadami Dirikhle, Izv. vuzov. Matematika, № 10, 91 – 93 (1975).

O. M. Mulyava, M. M. Sheremeta, Relative growth of Dirichlet series, Mat. Stud., 49, № 2, 158 – 164 (2018), https://doi.org/10.15330/ms.49.2.158-164

O. M. Mulyava, M. M. Sheremeta, Remarks to relative growth of entire Dirichlet series, Visnyk Lviv Univ. Ser. Mech., Math., № 87, 73 – 81 (2019).

Yu. M. Gal`, M. N. Sheremeta, O roste analiticheskikh v poluploskosti funkczij, zadanny`kh ryadami Dirikhle, Dokl. AN USSR, ser., № 12, 1065 – 1067 (1978).

J. F. Ritt, On certain points in the theory of Dirichlet series, Math. Amer. J., 50, 73 – 83 (1928), https://doi.org/10.2307/2370849

A. M. Gajsin, Oczenki rosta funkczij, predstavlenny`kh ryadami Dirikhle v poluploskosti, Mat. sb, 117, № 3, 412 – 424 (1982).

M. M. Sheremeta, Asimptoticheskoe povedenie czely`kh funkczij, zadanny`kh stepenny`mi ryadami i ryadami Dirikhle, Diss. . . . d-ra fiz.-mat. nauk. – Kiev (1987).

Yu. M. Gal`, O roste analiticheskikh funkczij, zadanny`kh absolyutno skhodyashhimisya v poluploskosti ryadami Dirikhle, Drogoby`ch (1980). – 30 с. – Dep. v VINITI, 1980, № 4080-80 Dep.

A. G. Azpeitia, A remark on the Ritt order of entire functions defined by Dirichlet series, Proc. Amer. Math. Soc., 12, 722 – 723 (1961), https://doi.org/10.2307/2034864

A. G. Azpeitia, On the lower linear type of entire functions defined by Dirichlet series, Bull. Unione Mat. Ital. A, 15, № 3, 635 – 638 (1978).

V. S. Bojchuk, O roste ryadov Dirikhle, absolyutno skhodyashhikhsya v poluploskosti, Mat. sb., Nauk. dumka, Kiev (1976), p. 238 – 240.

M. M. Sheremeta, S. I. Fedy`nyak, O proizvodnoj ryada Dirikhle, Sib. mat. zhurn., 39, № 1, 206 – 223 (1998).

Published
13.11.2020
How to Cite
MulyavaO. M., and Sheremeta M. M. “Relative Growth of Dirichlet Series With Different Abscissas of Absolute Convergence”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 11, Nov. 2020, pp. 1535-43, doi:10.37863/umzh.v72i11.6168.
Section
Research articles