Absolute Tauberian constants for Cesàro means

Abstract:

This paper is concerned with introducing two inequalities of the form $\sum \nolimits _{n = 0}^\infty {|{\tau _n}} - {a_n}| \leqq KA$ and $\sum \nolimits _{n = 0}^\infty {|{\tau _n}} - {a_n}| \leqq K’B$, where ${\tau _n} = C_n^{(k)} - C_{n - 1}^{(k)},C_n^{(k)}$ denote the Cesàro transform of order $k,K$ and $K’$ are absolute Tauberian constants, $A = \sum \nolimits _{n = 0}^\infty {|\Delta (n{a_n}} )| < \infty ,B = \sum \nolimits _{n = 0}^\infty {|\Delta ((1/n)\sum \nolimits _{v = 1}^{n - 1} {v{a_v}} } )| < \infty$ and $\Delta {u_k} = {u_k} - {u_{k + 1}}$. The constants $K,K’$ will be determined.