We prove that if \(X^{*}\) is strictly convex, a convex function \(f\) is coercive and b-Lipschitzian iff there exist two convex function sequences \(\{f_{n}\}_{n=1}^{\infty}\) and \(\{g_{n}\}_{n=1}^{\infty}\) such that (1) \(f_{n}\leq f_{n+1}\leq f\) and \(f\leq g_{n+1}\leq g_{n}\) for all integers \(n \geq 1\); (2) \(f_{n}\) and \(g_{n}\) are continuous and Gâteaux differentiable on \(X\); (3) \(f_n \to f\) and \({g_n \to f}\) uniformly on \(X\); (4) \(f_n\) and \(g_n\) are coercive and b-Lipschitzian. Moreover, we also prove that if \(X^{*}\) is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists \(m>0\) such that \(|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|\) and \(|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|\) whenever \(x, y \in X\) and \(n\in N\).

我们证明如果\（X ^ {*} \）是严格凸的，则凸函数\（f \）是强制的，并且如果存在两个凸函数序列 \（\ {f_ {n} \} _ { n = 1} ^ {\ infty} \）和\（\ {g_ {n} \} _ {n = 1} ^ {\ infty} \）这样（1）\（f_ {n} \ leq f_ { n + 1} \ leq f \）和\（f \ leq g_ {n + 1} \ leq g_ {n} \）对于所有整数\（n \ geq 1 \） ; （2）\（f_ {n} \）和\（g_ {n} \）是连续的，并且Gâteaux在\（X \）上是可区分的；（3）\（f_n \ to f \）和\（{g_n \ to f} \）统一放在\（X \）上；（4）\（f_n \）和\（g_n \）是强制性的，是b-Lipschitzian。此外，我们还证明，如果\（X ^ {*} \）是严格凸的，则凸函数f为Lipschitzian iff条件（1）-（3）为真，并且存在\（m> 0 \）使得\（| f_ {n}（x）-f_ {n}（y）| \ leq m \ | xy \ | \）和\（| g_ {n}（x）-g_ {n}（y）| \每次\（x，y \ in X \）和\（n \ in N \）时都应使用leq m \ | xy \ | \\）。