Absolute Tauberian constants for Cesàro means of a function

Abstract:

This paper is concerned with introducing two estimates of the forms $F \leqslant C{A_k}(\alpha ),F \leqslant D{B_k}(\alpha ),(\alpha > 0)$, where $F = \smallint _0^\infty {|d\{ f(\alpha x) - {\sigma _k}(x)\} |,{\sigma _k}(x)}$ denote the Cesàro transform of order k of the function $f(x) = \smallint _0^x {g(t)\;dt,g(t)}$ is a function of bounded variation in every finite interval of $t \geqslant 0,{A_k}(\alpha ),{B_k}(\alpha )$ are absolute Tauberian constants, $C = \smallint _0^\infty {|d\{ tg(t)\} | < \infty ,D = \smallint _0^\infty {|d\{ \phi (t)\} | < \infty } }$ and $\phi (t) = {t^{ - 1}}\smallint _0^t {ug(u)du}$. The constants ${A_k}(\alpha ),{B_k}(\alpha )$ will be determined.