On the derivative of Dirichlet series (in Ukrainian)

For an absolutely convergent in $(-\|,A)$ Dirichlet series $F(s) = \Sigma_{n=0}^{\|} a_n \text{e}^{s\lambda_n },\; s=\sigma+it$, $0 < \lambda_n\uparrow +\|\; (n\to \|)$, $A \in (-\|,+\|]$, let $M(\sigma ,F) = \sup\{ |F(\sigma +it)| : t \in \Bbb{R}\}$ and $S_1 (\sigma ,F) =\frac {M(\sigma ,F^\prime )}{M(\sigma ,F)}$, $\sigma < A$. Let $\Omega(A)$, $-\| < A\le +\|$ be the class of positive unlimited on $(-\|,A)$ functions $ \Phi$ for which the derivative $\Phi^\prime$ is a positive, continuous and increasing function on $(-\|,A)$ with limit $+\|$. Let $\Psi (x)= x-\frac{\Phi (x)}{\Phi^\prime(x)}$. We have the following assertion. Let $q \in (0,+\|)$, $A\in (-\|,+\|]$, and let $\Phi \in \Omega(A)$ be a function such that $\ln \Phi^\prime (\sigma) \le \frac{1}{q}\bigl(\Phi(\sigma)+(A-\sigma)\Phi^\prime(\sigma)\bigr)$, $\sigma_0\le\sigma < A. $ If for a Dirichlet series $F(s)$ we have $ n(t)\le t^q \;\;(t\ge t_{\circ})$ and $$\varlimsup_{\sigma \to A} \frac{\ln M(\sigma,F)}{\Phi (\sigma)}=1,$$ then $$ \varlimsup_{\sigma \to A} \frac{S_1(\sigma,F)}{\Phi^\prime(\Psi^{-1}(\sigma+q\beta(\sigma)))} \le 1 \le \varlimsup_{\sigma \to A} \frac{S_1(\sigma,F)}{\Phi^\prime(\sigma)},$$ where $\beta (\sigma) = \frac {\ln \Phi^\prime (\Psi^{-1} (\sigma))}{ \Phi^\prime (\Psi^{-1} (\sigma))}.$