On the rate of growth of entire functions of fast growth

2. Definitions. Notations and preparatory lemmas. NOTATION 1. exp =log ^ = x, exp# = log~x = exp(exp'-%) = log(logt~-%) (m = 0, ± 1 , ± 2 , • • • ). NOTATION 2. r r £«(*) n «pro *, Aw(«) n iog w *, E[-r](x) = »/A[r-i](»), Afr-riO*) = a?/£[r-.i](«), x = £M(y) « y = £[r](«) (f = 0, ± 1, ± 2 , • • • ). LEMMAS. 77^ functions expx, log#, E[r](x), AM(X), E^ ^) (m = 0, ± 1 , ± 2 , • • • ; r = 0, 1, 2, • • • ) all increase monotonically and we have