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

Abstract

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, NorthHolland Publishing Company, Amsterdam, 1991, Colloq. Math. Soc. János Bolyai 58 (1991) 355-360] on Bleimann-Butzer-Hahn operators L n . Our main result states that for each locally bounded positive function h there exists a continuous positive function f defined on [0, ∞) with L n f → f (n → ∞), pointwise on [0, ∞), such that lim sup x→+∞ f (x) h(x) = +∞.Moreover we construct an explicit counterexample function to Hermann's conjecture.