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

O. M. Mulyava

doi:10.30970/ms.55.1.44-50

O. M. Mulyava Kyiv National University of Food Technologies, Kyiv, Ukraine

DOI: https://doi.org/10.30970/ms.55.1.44-50

Анотація

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.

Посилання

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