On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases

It is proved that if a convex density-like differential basis $B$ is centered and invariant with respect to translations and homotheties, then the integral means of a nonnegative integrable function with respect to $B$ can boundedly diverge only on a set of measure zero (this generalizes a theorem of Guzmán and Menarguez); it is established that both translation and homothety invariances are necessary.