Sequences Behavior and Applications

Abstract

Let j be a fixed integer. Let $\{x_n\}$ be a sequence of complex numbers and let $\Delta x_j = x_{j+1} - x_j.$ Let $0< \alpha <1,$ first, it is shown that (a) $|x_n| = O(e^{\epsilon n^{\alpha }}),$ as $n \rightarrow \infty$ for all $\epsilon > 0,$ and (b) $|\Delta ^n x_j|^{\frac{1}{n}} \rightarrow 0$ as $n \rightarrow \infty ,$ if and only if there exists a positive constant R such that $\Vert \Delta ^{n} x_j\Vert \le R^n n^{n(1-\frac{1}{\alpha })}.$ 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 Cesàro means $M_n(x_n)$ and retaining the conclusion. Applications to the orbits of operators are given. This helps to elucidate and improve former results of Mbekhta–Zemánek, Atzmon and others.