Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series

Let $A\in(-\infty,+\infty]$, $\Phi:[a,A)\to\mathbb{R}$ be a continuous function such that $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$ for every $x\in\mathbb{R}$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ and $\Gamma(x)=(\widetilde{\Phi}(x)-\ln x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, and $F(s)=\sum\limits\limits_{n=0}^\infty a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximal term $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and central index $\nu(\sigma,F)=\max\{n\ge0:|a_n|e^{\sigma\lambda_n}=\mu(\sigma,F)\}$ are defined for all $\sigma