Approximation and shape preserving properties of the nonlinear Favardsza-Szász-Mirakjan operator of max-product kind

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Starting from the study of the Shepard nonlinear operator of max-prod type in [6], [7], in the book [8], Open Problem 5.5.4, pp. 324-326, the Favard-Sz?sz-Mirakjan max-prod type operator is introduced and the question of the approximation order by this operator is raised. In the recent paper [1], by using a pretty complicated method to this open question an answer is given by obtaining an upper pointwise estimate of the approximation error of the form C?1(f;?x/?n) (with an unexplicit absolute constant C>0) and the question of improving the order of approximation ?1(f;?x/?n) is raised. The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages : it produces an explicit constant in front of ?1(f;?x/?n) and it can easily be extended to other max-prod operators of Bernstein type. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ?1(f;?) cannot be improved. However, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order ?1 (f;1/n) is obtained. Finally, some shape preserving properties are obtained.