In this paper we investigate the asymptotic relation between maximum moduli of a class of functions analytic on the unit disc and their partial sums, i.e\. we formulate the problem of best $\lambda$-approximations. We give a solution of the best $\lambda$-approximation for analytic functions of rapid growth on the unit disc such as, for example, is the Hardy-Ramanujan generating partition function. Using Ingham Tauberian Theorem we give some interesting applications. Results for functions of medium growth and for entire functions of finite order are also quoted. In growth-measuring an essential role is played by Karamata's class of regularly varying functions.