The growth of entire Dirichlet series in terms of generalized orders

Let $\alpha$ be a continuous function which increases to $+\infty$ on an infinite interval of the form $[x_0,+\infty)$. A necessary and sufficient condition is found on a sequence $(\lambda_n)_{n=0}^\infty$ increasing to $+\infty$ which ensures that for each Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, which is absolutely convergent in $\mathbb{C}$ the following relation holds:
$$ \varlimsup_{\sigma\to+\infty}\frac{\alpha(\ln M(\sigma,F))}{\sigma}=\varlimsup_{\sigma\to+\infty}\frac{\alpha(\ln\mu(\sigma,F))}{\sigma}, $$
where $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re} s=\sigma\}$ and ${\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}\colon n\geqslant 0\}}$ are the maximum modulus and maximum term of the series, respectively.
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