On a general class of geometric constants and normal structure in Banach spaces

In this paper, we establish the lower bounds for the weakly convergent sequence coefficient WCS(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {WCS}(X)$$\end{document} of a Banach space X, in terms of the family of geometric constants C lambda,p(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\lambda ,p}(X)$$\end{document}, the coefficient of weak orthogonality mu(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (X)$$\end{document}, and the generalized Garc & iacute;a-Falset coefficient R(1, X). By means of these bounds, we identify some geometrical properties implying normal structure. Moreover, some sufficient conditions which imply uniform normal structure are presented. Our results significantly generalize and improve many known results in the literature.