On the theorem of Frullani

Juan Arias-de-Reyna
1990 Proceedings of the American Mathematical Society  
We prove that, for every function /: R+ -> C such that (f(ax)f(bx))/x is Denjoy-Perron integrable on [0,+co) for every pair of positive real numbers a , b , there exists a constant A (depending only on the values of f(t) in the neighborhood of 0 and -t-co ) such that rnax)-fmdx=A l0« Jo x b To prove this assertion, we identify a Denjoy-Perron integrable function /: R -» C with a distribution. In this way, we obtain the main result of this paper: The value at 0 (in Lojasiewicz sense) of the
more » ... er transform of the distribution f is the Denjoy-Perron integral of /. Assuming the Continuum Hypothesis, we construct an example of a non-Lebesgue measurable function that satisfies the hypotheses of the first theorem.
doi:10.1090/s0002-9939-1990-1007485-4 fatcat:ugixkmhiufedpbjlh7muu7mx5y