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On the theorem of Frullani

1990
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Proceedings of the American Mathematical Society
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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

doi:10.1090/s0002-9939-1990-1007485-4
fatcat:ugixkmhiufedpbjlh7muu7mx5y