In this paper, we have derived certain classical inequalities, namely Young's, Hölder's, Minkowski's and the Hermite–Hadamard inequalities for a pseudo-integral (also known as g-integral). For Young's, Hölder's and Minkowski's inequalities, the cases p > 1 and p < 1 , p ≠ 0 , have been covered. Moreover, in the case of the Hermite–Hadamard inequality, the refinement has also been proved and, as a special case, a g-analogue of a geometric-logarithmic-arithmetic inequality has been deduced.