Banas–Hajnosz–Wedrychowicz type modulus of convexity and normal structure in Banach spaces

In this paper, we present some sufficient conditions for which the Banach space X has uniform normal structure in terms of the Banas–Hajnosz–Wedrychowicz type modulus of convexity SYX(ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SY_X(\epsilon )$$\end{document}, the coefficient of weak orthogonality ω(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (X)$$\end{document} and the Domínguez–Benavides coefficient R(1, X). Some known results are improved and strengthened.