Ukrainian Mathematical Journal ( IF 0.518 ) Pub Date : 2021-06-10 , DOI: 10.1007/s11253-021-01887-1
O. M. Mulyava, M. M. Sheremeta

We study the growth of a Dirichlet series $$F(s)={\sum}_{n=1}^{\infty }{f}_n\exp \left\{s{\uplambda}_n\right\}$$ with zero abscissa of absolute convergence with respect to the entire Dirichlet series $$G(s)={\sum}_{n=1}^{\infty }{g}_n\exp \left\{s{\uplambda}_n\right\}$$ by using generalized quantities of an order $${\upvarrho}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\sup}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)}$$ and a lower order $${\uplambda}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\operatorname{inf}}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)},$$ where $${M}_F\left(\sigma \right)=\sup \left\{\left|F\left(\sigma + it\right)\right|:t\in \mathrm{\mathbb{R}}\right\},{M}_G^{-1}(x)$$ is the function inverse to MG(σ), and β is a positive function increasing to +∞.

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