On entire Dirichlet series similar to Hadamard compositions
Abstract
A function F(s)=∑∞n=1anexp{sλn} with 0≤λn↑+∞ is called the Hadamard composition of the genus m≥1 of functions Fj(s)=∑∞n=1an,jexp{sλn} if an=P(an,1,...,an,p), where
P(x1,...,xp)=∑k1+⋯+kp=mck1...kpxk11⋅...⋅xkpp is a homogeneous polynomial of degree m≥1. Let M(σ,F)=sup and functions \alpha,\,\beta be positive continuous and increasing to +\infty on
[x_0, +\infty). To characterize the growth of the function M(\sigma,F), we use generalized order \varrho_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\alpha(\ln\,M(\sigma,F))}{\beta(\sigma)}, generalized type
T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}
and membership in the convergence class defined by the condition
\displaystyle \int_{\sigma_0}^{\infty}\frac{\ln\,M(\sigma,F)}{\sigma\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}d\sigma<+\infty.
Assuming the functions \alpha, \beta and \alpha^{-1}(c\beta(\ln\,x)) are slowly increasing for each c\in (0,+\infty) and \ln\,n=O(\lambda_n) as n\to \infty, it is proved, for example, that if the functions F_j have the same generalized order \varrho_{\alpha,\beta}[F_j]=\varrho\in (0,+\infty) and the types T_{\alpha,\beta}[F_j]=T_j\in [0,+\infty), c_{m0...0}=c\not=0, |a_{n,1}|>0 and |a_{n,j}|= o(|a_{n,1}|) as n\to\infty for 2\le j\le p, and F is the Hadamard composition of genus
m\ge 1 of the functions F_j then \varrho_{\alpha,\beta}[F]=\varrho and
\displaystyle T_{\alpha,\beta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1T_1+...+k_pT_p).
It is proved also that F belongs to the generalized convergence class if and only if
all functions F_j belong to the same convergence class.
References
Mulyava O.M., Sheremeta M.M. Compositions of Dirichlet series similar to the Hadamard compositions, and convergence classes // Mat. Stud. – 2019. – V. 51, №1. – P. 25–34.
Hadamard J. Theoreme sur le series entieres // Acta math. – 1899. – Bd. 22. – S. 55–63.
Hadamard J. La serie de Taylor et son prolongement analitique // Scientia phys.-math. – 1901. – №12. – P. 43–62.
Bieberbach L. Analytische Fortzetzung. – Berlin, 1955.
Calys E.G. A note on the order and type of integral functions // Riv. Mat. Univer. Parma (2). – 1964. – V. 5. – P. 133–137.
Kulyavetc’ L.V., Mulyava O.M. On the growth of a class of entire Dirichlet series // Carpathian Math. Publ. – 2014. – V. 6, №2. – P. 300–309.(in Ukrainian)
Sheremeta M.M. Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion // Izv. Vyssh. Uchebn. Zaved. Mat. – 1967 – №2. – P. 100–108.(in Russian)
Sheremeta M.M. Entire Dirichlet series. – Kyiv: ISDO. – 1993, 168 p. (in Ukrainian)
Leont’ev A.F. Series of exponents. – M.: Nauka, 1976. – 536 p. (in Russian)
Ritt J.F. On certain points in the theory of Dirichlet series // Amer. Math. J. – 1928. – V. 50. – P. 73–83.
Azpeitia A.G. On the lower linear type of entire functions defined by Dirichlet series // Bull. Unione Mat. Ital. – 1978. – V. A15, №3. – P. 635–638.
Sheremeta M.M. On two classes of positive functions and belonging to them of main characteristics of entire functions // Mat. Stud. – 2003. – V. 19, №1. – P. 75–82.
Sheremeta M.M., Fedynyak S.I. On the derivative of a Dirichlet series // Sibirsk. mat. journ. – 1998. – V. 39, №1. – P. 206–223. (in Russian)
Mulyava O.M., Sheremeta M.M. Convergence classes of analytic functions. – Kyiv: Publ. Lira K., 2020. – 196 p.
Valiron G. General theory of integral functions. – Toulouse, 1923.
Kamthan P.K. A theorem of step functions. II // Instambul univ. fen. fac. mecm. A. – 1963. – V. 28. – P. 65–69.
Mulyava O.M. On convergence classes of Dirichlet series // Ukr. Math. Journ. – 1999. – V. 51, №1. – P. 1485–1494. (in Ukrainian)
Copyright (c) 2023 O.M. Mulyava, M. M. Sheremeta

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.