In this work we show that the gravity Lagrangian at relatively low curvatures in both metric and Palatini formalisms is a bounded function that can only depart from the linearity within the limits defined by well-known functions. We obtain those functions by analyzing a set of inequalities that any theory must satisfy in order to be compatible with laboratory and solar system observational constraints. This result implies that the recently suggested f(R)gravity theories with nonlinear terms that dominate at low curvatures are incompatible with observations and, therefore, cannot represent a valid mechanism to justify the cosmic speedup.