1. The purpose of this paper is to present a general integral inequality concerning subadditive functions and to make applications of this inequality. The applications pertain to relations among integrals involving first and second differences of LP functions. The finiteness of some of the integrals is connected with generalized Lipschitz conditions and with the existence of fractional derivatives. These facts are exploited to obtain both new and known theorems. Finally we show that in some cases the finiteness of the integral is not affected by interchanging the first and second differences of the function. We say the positive measurable function 4 is subadditive on the interval (0, A), 0 1, of qP(u)/ua with respect to the infinite measure du/u does not exceed a constant multiple of the L norm of this function with respect to the same measure. Here a is any real number.