Extreme values and entire functions of finite order

For an arbitrary entire function g   of finite order and with non-negative Taylor coefficients, we shall prove that its restriction to the positive part of real axis belongs to de Haan's class ΓΓ. It follows that the laws F1(x)=1-1/g(x)F1(x)=1-1/g(x) and F2(x)=∫-∞xdt/g(|t|)/C are in the domain of attraction of the double exponential law with explicitly given norming constants.