This paper generalizes the linear sector in the classical absolute stability theory to a sector bounded by concave/convex functions. This generalization allows more flexible or more specific description of the nonlinearity and will thus reduce the conservatism in the estimation of the domain of attraction. We introduce the notions of generalized sector and absolute contractive invariance for estimating the domain of attraction of the origin. Necessary and sufficient conditions are identified under which an ellipsoid is absolutely contractively invariant. In the case that the sector is bounded by piecewise linear concave/convex functions, these conditions can be exactly stated as linear matrix inequalities. Moreover, if we have a set of absolutely contractively invariant (ACI) ellipsoids, then their convex hull is also ACI and inside the domain of attraction. We also present optimization technique to maximize the absolutely contractively invariant ellipsoids for the purpose of estimating the domain of attraction. The effectiveness of the proposed method is illustrated with examples.