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The concern of this paper is a recent generalization L n ( f (t 1 , t 2 ); x, y) for the operators of Bleimann, Butzer, and Hahn in two variables which is distinct from a tensor product. We present the complete asymptotic expansion for the operators L n as n tends to infinity. The result is in a form convenient for applications. All coefficients of n &k (k=1, 2, ...) are calculated explicitly in terms of Stirling numbers of the first and second kind. As a special case we obtain a Voronovskaja-type theorem for the operators L n .

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We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Raşa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, $$q\in \mathbb N$$ q ∈ N . In particular, 1-monotone functions are nondecreasing functions, and 2-monotone functions are convex functions. In general, q-monotone functions on [0, 1], for $$q\ge 2$$ q ≥ 2 , possess a $$(q-2)$$ ( q - 2 ) nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.

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