On the question of representing entire functions by exponential series

It was established by the author (RZhMat., 1966, 4B 107) that any entire function $F(z)$ of finite order may be represented in the whole plane by a Dirichlet series
$$ F(z)=\sum_{k=1}^\infty A_ke^{|\lambda_k|z}. $$

It is established that for suitable choice of the sequence $\{\lambda_k\}$ the expression $\sum_{k=1}^\infty|A_k|e^{|\lambda_k|r}$ has, for large $r$, the upper bounds
1) $\exp r^{\rho+\varepsilon}$ $\forall\,\varepsilon>0$, if $F(z)$ has order $\rho>1$;
2) $\exp(\sigma+\varepsilon)r^\rho$ $\forall\,\varepsilon>0$, if $F(z)$ has order $\rho>1$ and finite type $\sigma$.
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