The duality between the uniform smoothness of the convex function \(f\) and the uniform convexity of its conjugate \(g\) is studied in the framework of Banach spaces in metric duality. Several characterizations of these notions are given using the subdifferentials of \(f\) and \(g.\) The paper is much related to reviewer's article [J. Math. Anal. Appl. 95, 344-374 (1983; Zbl 0519.49010)]. The reflexivity of the space and the interiority condition used in some implications of that paper are dropped and some proofs are simplified. Some results concerning maximal monotone operators are also obtained.