An answer to Hermann's conjecture on Bleimann–Butzer–Hahn operators

We give a negative answer to the conjecture of Hermann [On the operator of Bleimann, Butzer and Hahn, in: J. Szabados, K. Tandori (Eds.), Approximation Theory, Proc. Conf., Kecskemét/Hung., 1990, North-Holland Publishing Company, Amsterdam, 1991, Colloq. Math. Soc. János Bolyai 58 (1991) 355–360] on Bleimann–Butzer–Hahn operators LnLn. Our main result states that for each locally bounded positive function h there exists a continuous positive function f   defined on [0,∞)[0,∞) with Lnf→f(n→∞)Lnf→f(n→∞), pointwise on [0,∞)[0,∞), such thatlimsupx→+∞f(x)h(x)=+∞.Moreover we construct an explicit counterexample function to Hermann's conjecture.