We consider certain subfamilies, of the family of univalent functions in the open unit disk, defined by means of sufficient coefficient conditions for univalency. This article is devoted to studying the problem of the well-known conjecture of Zalcman consisting of a generalized coefficient functional, the so-called generalized Zalcman conjecture problem, for functions belonging to those subfamilies. We estimate the bounds associated with the generalized coefficient functional and show that the estimates are sharp.2010 Mathematics Subject Classification. Primary: 30C45, 30C55; Secondary: 30C50.