Growth estimates for a Dirichlet series and its derivative

  • S.I. Fedynyak Ukrainian Catholic University, Lviv, Ukraine
  • P.V. Filevych Lviv Polytechnic National University, Lviv, Ukraine
Keywords: analytic function, maximum modulus, maximum modulus point, zero set

Abstract

Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have
$x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series absolutely convergent in the half-plane $\operatorname{Re}s<A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma<A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality
$$
\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F')}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1
$$
holds, and this inequality is sharp. % Abstract (in English)

References

M.N. Sheremeta, S.I. Fedynyak, On the derivative of a Dirichlet series, Sib. Mat. Zh., 39 (1998), No1, 206–223. (in Russian) English translation in: Siberian Math. J., 39 (1998), №1, 181–197.

S.I. Fedynyak, On maximum modulus points and zero set for an entire function, Mat. Stud., 30 (2008), №2, 169–172.

M.A. Evgrafov, Asymptotic estimates and entire functions, Moscow: Nauka, 1979. (in Russian)

T.Ya. Hlova, P.V. Filevych, Generalized types of the growth of Dirichlet series, Carpathian Math. Publ., 7 (2015), №2, 172–187.

S.I. Fedynyak, P.V. Filevych, Distance between a maximum modulus point and zero set of an analytic function, Mat. Stud., 52 (2019), №1, 10–23.

A.F. Leont’ev, Series of exponents, Moscow: Nauka, 1976. (in Russian)

P.V. Filevych, On relations between the abscissa of convergence and the abscissa of absolute convergence of random Dirichlet series, Mat. Stud., 20 (2003), №1, 33–39.

P.V. Filevich, On Valiron’s theorem on the relations between the maximum modulus and the maximal term of an entire Dirichlet series, Izv. Vyssh. Uchebn. Zaved. Mat., (2004), №4, 66–72. (in Russian) English translation in: Russian Math., 48 (2004), №4, 63–69.

P.V. Filevich, On influence of the arguments of coefficients of a power series expansion of an entire function on the growth of the maximum of its modulus, Sib. Mat. Zh., 44 (2003), №3, 674–685. (in Russian) English translation in: Siberian Math. J., 44 (2003), №3, 529–538.

T.Ya. Hlova, P.V. Filevych, The growth of analytic functions in the terms of generalized types, J. Lviv Politech. Nat. Univ., Physical and mathematical sciences, (2014), №804, 75–83. (in Ukrainian)

Published
2020-03-17
How to Cite
Fedynyak, S., & Filevych, P. (2020). Growth estimates for a Dirichlet series and its derivative. Matematychni Studii, 53(1), 3-12. https://doi.org/10.30970/ms.53.1.3-12
Section
Articles