Increasing Self-Described Sequences

We proceed with our study of increasing self-described sequences F, beginning with 1 and defined by a functional equation In [1] we exhibited the simple solution f (t)=Ct, for some (0,1), of the associated functional-differential equation and we proved that provided <2/(2+d()), where we have the asymtotic equivalence F(m)~ Cm.In the present paper we show that this last result is optimal, in the sense that the self-described sequence defined by |F^–1(m)|=F(m)², that is for which the boundary case =2/(2+d())(=1/2) holds, does not satisfy F(m) ~ Cm. We also show that the m-th term F(m) of a sequence F for which the boundary case holds is nevertheless of asymptotic order m.Then we investigate the behaviour of self-described sequences F when lies beyond the boundary case. In [1] we established the estimates when is the unique fixed point of a certain associated function. We were only able to prove in general that the latter holds when does not lie beyond the boundary case, however. In the present paper we prove that whenever is the unique fixed point of this function, and in addition we obtain estimates more precise than (*). This applies for instance to the sequence defined by that is