Riemann derivatives and general integrals

Abstract

Sargent and later Bullen and Mukhopadhyay obtained a definition of absolutely continuous functions, functions, that is related to kth Peano derivatives. The generalised notions of ACkG*, [ACkG*], ACkG* above, etcetera functions led Bullen and Mukhopadhyay to define certain general integrals of the kth order.The present work is concerned with a further simplification of the definitions of such functions by the use of divided differences but still retaining similar fundamental properties. These concepts lead to the introduction of Denjoy and Ridder type integrals which are shown to be equivalent to a Perron type integral that corresponds to kth Riemann* derivatives. All three of these integrals are shown to be equivalent to the three integrals of Bullen and Mukhopadhyay.

Integration by parts for some general integrals

“…Since D k -and V k -integrals of De Sarkar and Das [14] and of Bullen and Mukhopadhyay [4] are equivalent to the P k -integral, Theorem 1 above also provides an integration by parts formula for each of these integrals.…”

“…If / is P-integrable on [a,b] then F(x) = (P) f* f, and if G is of bounded variation, then fG is P -integrable and If -TO is a P k -major function of -/, then m is called a P k -minor function of / on [a,b]. If -o o < inf{M(6)} = sup{m(6)} < +oo, then / is P k -integrable on [a,b] and the common value is called the P k -integral of / on [a,b], and is denoted by Following Bergin [1] and Remark 6 of De Sarkar and Das [14], we can say that D k~1 M is a (k -l)-majorant and D k~1 m is a (k -l)-minorant of / on [a,b] and the finite common value…”

Generalised Hölderian functions

“…For background and further information on BV».and AC* functions, we refer the reader to Russell [13,14], De Sarkar and Das [4,5,6,7], De Sarkar, Das and Lahiri [8] and Das and Das [3].…”