On an estimation of R-type of entire Dirichlet series and its exactness

  • T.Ya. Hlova Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine
  • P.V. Filevych Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Keywords: entire Dirichlet series, maximum modulus, maximum term, R-type
Published online: 2013-12-30


Let $(\lambda_n)$ be a nonnegative sequence, increasing to $+\infty$, $\tau=\limsup\limits_{n\to\infty}\frac{\ln n}{\lambda_n}$, and $\rho$ be a positive number. It follows from a classical theorem of G. Valiron that for every Dirichlet series of the form $F(s)=\sum a_ne^{s\lambda_n}$ we have

$$\limsup_{\sigma\to+\infty}\frac{\ln \sup\{|F(s)|:\,\text{Re}\, s=\sigma\}}{e^{\rho\sigma}}\le e^{\rho\tau} \limsup_{n\to\infty}\frac{\lambda_n}{e\rho}|a_n|^\frac{\rho}{\lambda_n}.$$

The exactness of this estimation is proved in the paper.

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How to Cite
HlovaT., FilevychP. On an Estimation of R-Type of Entire Dirichlet Series and Its Exactness. Carpathian Math. Publ. 2013, 5 (2), 208-216.