On behaviour of the maximal term of Dirichlet series derivative

Let $\mu(\sigma,F)$ be the maximal term of Dirichlet series $F(s)=\sum_{n=0}^{\|}a_n\exp\{s\lambda_n\}$, $k s=\sigma+it$, $ 0\le\lambda_n\uparrow+\|$, with the abscissa of absolute convergence $A\in(-\|,+\|]$. Conditions on positive convex function $\Phi$ on $(-\|,\,A)$ are studied in order that the relation $\ln\,\mu(\sigma,F)=(1+o(1))\Phi(\sigma),\, \sigma\uparrow A$, imply the relation $\mu(\sigma,F')/\mu(\sigma,F)= (1+o(1))\Phi'(\sigma)$, $\sigma\uparrow A$.