These classes of functions can be characterized in terms of the rate of growth of Taylor coefficients:
MR0218567

Šeremeta, M. N.
Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion. (Russian)
Izv. Vysš. Učebn. Zaved. Matematika 1967 1967 no. 2 (57), 100–108.

MR0330452 Šeremeta, M. N. The coefficients of the power series expansion of entire functions. (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen. No. 16 (1972), 41–44, 216.

If you prefer something using Latin alphabet, there is an old book

O. Blumenthal, Principes de la theorie des fonctions entieres d'ordre infini, Paris,
Gauthier-Vilars, 1910.