| In this work, we give explicit upper bounds of central moments for fivekinds of classical probabilistic operators which are Bernstein operator, Sza′szoperator, Baskakov operator, Post-Widder operator and Meyer-K¨onig andZeller operator. As an expansion, we also give explicit upper bound of centralmoment for Feller operator. Finally, we give explicit upper bound of centralmoment for q-Bernstein operator(0 < q < 1) and the existence of upper boundof central moment for q-Sza′sz operator(0 < q < 1) defined by Aral; both ofthem are extensively studied.In particular, in order to obtain the desired result for Bernstein operator,Post-Widder operator and Feller operator, we derive general expressions oftheir central moments. Suppose l∈N+, as for Bernstein operator, we can usea polynomial inφ2(x) of degree equal to l to express 2l order central momentand the product of a polynomial inφ2(x) of degree equal to l and (φ2(x)) toexpress 2l + 1 order central moment; the cases of Post-Widder operator aresimilar to those of Bernstein operator; as for Feller operator, we can use a com-bination ofφ2lφ<sup>2i(x)[(φ2(x)) ]2i, and a combination ofφ2lφ<sup>2i(x)[(φ2(x)) ]2i+1,i = 0,1,...,l?1, to express its 2l and 2l+1 order central moment, respectively.?(x) is the corresponding step-weight function for these operators. We refer-ence the results of Guo and Qi [Applied Mathematics Letters, 2007] and obtaina trivial explicit upper bound of central moment for Meyer-Ko¨nig and Zelleroperator. In order to obtain the result for q-Bernstein operator(0 < q < 1)we reference the work of Mahmudov [Numer. Algor., 2010]. We obtain theexistence of upper bound of central moments of classical Sza′sz operator fromthat of q-Sza′sz operator(0 < q < 1), as q tends to 1 from the left side.The method of this work can be applied in deriving upper bounds ofcentral moments for other similar operators and the results of this work canbe used to study approximation properties of corresponding operators.
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