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, North-Holland 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.