We establish sufficient and necessary condition for $\int_0^{+∞} e^{-xp} lnμ(x,F) < +∞, p > 0,$ for a positive functional series of the form $F(x) = \sum_{n=0}^{+∞} α_n e^{xλ_n + \tau (x)β_n}, α_n ≥ 0, (n ≥ 0),$ convergent for $x ≥ 0$, where $\tau (x)$ is positive increase, $(λ_n), (β_n)$ are an positive sequences, $μ(x,F) = max \{α_n e^{xλ_n + \tau (x)β_n} : n ≥ 0\}.$ 

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