Summary: In \(\mathbb{R}^n\), given \(\gamma\in[0,n)\) and \(p\in(1,n/\gamma)\), it is well known that \(w^q\in A^r\), with \(1/q= 1/p-\gamma/n\) and \(\displaystyle{r=1+q{p-1\over p}}\), is a necessary and sufficient condition for the boundedness of the maximal fractional operator \(M_\gamma\) between \(L^p(w^p)\) and \(L^q(w^q)\) spaces. In this work, we study the dependence of the operator norm on the constant of the \(A_r\) condition. The result extends that obtained by S. Buckley for the Hardy-Littlewood maximal function (i.e., \(\gamma=0\)).