The energy $E$ of the lowest discrete level of a quantum-mechanical system is considered as a function of a parameter $\lambda$, that occurs linearly in the energy operator. An inequality that generalizes the well-known convexity property of the function $E(\lambda)$ is derived. The application of the generalized convexity property is illustrated by the example of the calculation of bounds for the total energies and the energies of the electron-nucleus interaction in the ground state for two-electron atoms.