On the Monotonicity of Positive Linear Operators Original Research Article

We provide sufficient conditions for a sequence of positive linear approximation operators,Ln(f, x), converging tof(x) from above to imply the convexity off. We show that, for the convolution operators of Feller type,Kn(f, x), generated by a sequence of iid random variables taking values in an intervalI, having a finite moment generating function, the inequalitiesKn(f, x)⩾f(x) (x∈I,n⩾1) are necessary and sufficient conditions forfto be convex. This provides a converse of a well-known result of R. A. Khan (Acta. Math. Acad. Sci. Hungar.39(1980), 193–203). It contains, as a special case, the corresponding result for the Bernstein polynomials and extends two results obtained for bounded continuous functions by Horova for Szász and Baskakov operators. As examples, similar results are also provided for the beta, Meyer-König Zeller, Picard, and Bleiman, Butzer, and Hahn operators.