Let \(f^*_ w(t)=\inf \{\lambda >0;w(\{x\in {\mathbb{R}}^ n;| f(x)| >\ell \})\leq t\}\) be the non-decreasing rearrangement of f with respect to a weight w, Mf be the Hardy-Littlewood maximal function of f, and \(M_ qf=M(| f|^ q)^{1/q}.\) The author shows for \(w\in A_{\infty}\). Muckenhoupt's weight class, \[ (Tf)^*_ w(t)\leq C(M_ qf)^*_ w(\gamma t)+(Tf)^*_ w(2t),\quad t>0(0<\gamma <1), \] when T is one of the Littlewood-Paley g-function, g(f), the Lusin area function, S(f), and the Littlewood-Paley function \(g^*_{\lambda}(f)\). In the above, \(q=1\) for \(T=g\) or S, and \(q=2/\lambda\) for \(T=g^*_{\lambda}(1<\lambda <2)\), i.e., the critical index for which weak type inequalities hold. He gives some consequences of iterating the above inequalities, along the line of the method in \textit{R. Bagby} and \textit{D. Kurtz}, Trans. Am. Math. Soc. 293, 71-81 (1986; Zbl 0585.42018). Several other applications are given.