On Tauberian conditions for the Hölder integrability method

Let f be a real-valued continuous function on [0,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}. Although every convergent integral s(x)=integral 0xf(t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s(x)=\int _0<^>xf(t)dt$$\end{document} is (H, 1) summable, the converse-deducing convergence from (H, 1) summability-requires Tauberian conditions. In this study, we demonstrate that the slowly decreasing condition (SD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{D}$$\end{document}), which is more general than convergence, is sufficient to establish the convergence of integrals that are (H, 1) and (H, 2) summable. Our results extend the theorems in [1, 2] and introduce weaker Tauberian conditions applicable to (H, k) summability methods.