Let j be a fixed integer. Let {xn} be a sequence of complex numbers and let ∆xj = xj+1 -xj. Let 0 < α < 1, first, it is shown that (a) |xn| = O(eϵnα) as n → ∞ for all ϵ > 0, and (b) |Δnxj |1/n→ 0 as n → ∞, if and only if there exists a positive constant R such that ||Δnxj || ≤ Rnnn(1- 1/a). Second, we present an extended version of our result [3], where the condition on the growth of the sequence is replaced by the same condition on the Cesaro means Mn(xn) and retaining the conclusion. Applications to the orbits of operators are given.This helps to elucidate and improve former results of Mbekhta-Zemánek, Atzmon