Monotone functions of several variables and the multidimensional second mean-value theorem (Q1896811)

The paper is devoted to the proof of the second mean value theorem in multidimensional setting and in two different forms: one in the sense of Bonnet, which was first proved by \textit{G. H. Hardy} [Proc. Lond. Math. Soc., II. Ser. 15, 72-88 (1916; J.-buch F.d.M. 46, 493)] under the assumption of Riemann integrability of the function \(f\) in question; another in the sense of du Bois-Reymond. The present author only assumes that \(f\) is integrable in Lebesgue's sense. Four lemmas are proved on properties of multidimensional differences, and three lemmas on monotone functions in several variables in the sense of W. H. Young. Two more lemmas give precise formulations to the multidimensional Abel transformation (or multiple summation by parts), which has been a folklore since a long time.