A class of distribution functions F(x), in the domain of attraction for maxima of the Gumbel law \(\Lambda (x)=\exp (-\exp (-x))\), \(x\in R\), is considered in particular, relevant elements of this class being the normal and gamma distributions. Applying a technique similar in spirit to that one used by P. Hall [J. Appl. Probab. 16, 433-439 (1979; Zbl 0403.60024)], we develop uniform upper and lower bounds for \[ \sup_{x\in R}| F^ n(a_ nx+b_ n)-\Lambda (x)| \] for a suitable choice of attraction coefficients \(\{a_ n\}_{n\geq 1}\) \((a_ n>0)\), and \(\{b_ n\}_{n\geq 1}\). The bounds obtained in a normal context compare favourably with the ones obtained in Hall’s paper. A few unsolved points related to gamma distributions with shape parameter smaller than one are emphasized.