Summary: We show the existence of an operator of the type
, such that preserves linear functions: and , , where is not necessarily a concave function on . The generalized Bernstein-type operators have been introduced and studied in [I. Gavrea and D. H. Mache, Generalization of Bernstein-type approximation methods. Müller, Manfred W. (ed.) et al., Approximation theory. Proceedings of the 1st international Dortmund meeting IDoMAT 95 held in Witten, Germany, March 13–17, 1995. Berlin: Akademie Verlag. Math. Res. 86, 115–126 (1995; Zbl 1005.41500)] and [M. Felten, J. Approximation Theory 94, No. 3, 396–419 (1998; Zbl 0913.41008)], respectively. In this paper we establish direct and converse theorems for the above-mentioned operators under more general conditions concerning the weight functions.