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In 1976, Donald Cartwright and John McMullen characterized axiomatically theRiemann-Liouvile fractional integral in a paper that was published in 1978. Themotivation for their work was to answer affirmatively to a conjecture stated byJ. S. Lew a few years before, in 1972. Essentially, their ``Cartwright-McMullentheorem in fractional calculus'' proved that the Riemann-Liouville fractionalintegral is the only continuous extension of the usual integral operator topositive real orders, in such a way that the Index Law holds. In this paper, wepropose an analogous result for the uniqueness of the extension of theStieltjes integral operator, in the case of a smooth integrator.

An extension of the Cartwright-McMullen theorem in fractional calculus for the smooth Stieltjes case