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Bede et al. (B. Bede, L. Coroianu, and S. G. Gal, Approximation and shape preserving properties of the nonlinear Meyer-König and Zeller operator of max-product kind , Numer. Funct. Anal. Optim. 31 (2010), no. 3, 232–253) defined the max-product Meyer-König and Zeller operator. They examined the approximation and shape preserving properties of this operator, and they found the order of approximation to be y ( 1 − y ) m \frac{\sqrt{y}(1-y)}{\sqrt{m}} by the modulus of continuity and claimed that this order of approximation could only be improved in certain subclasses of the functions. In contrast to this claim, we demonstrate that we can obtain a better order of approximation without reducing the function class (by the classical modulus of continuity). We find the degree of approximation to be ( 1 − y ) y 1 α m 1 − 1 α \frac{(1-y){y}^{\tfrac{1}{\alpha }}}{{m}^{1-\tfrac{1}{\alpha }}} , α = 2 , 3 , … \alpha =2,3,\ldots . Since 1 − 1 α 1-\frac{1}{\alpha } tends to 1 for enough big α \alpha , we improve this degree of approximation.

Nonlinear q-Bernstein operator of max-product kind was introduced and its approximation order was examined, and the order of approximation was found to be by Duman in. In this paper, we obtained a better order of approximation for this operator.

Using maximum instead of sum, nonlinear Bernstein operator of maximum product kind is introduced by Bede et al. [2]. The present paper deals with the approximation processes for this operator. The order of approximation for this operator to the function f, can be found in [4] by means of the classical modulus of continuity. Also, in [4], it was indicated that the order of approximation of this operator to the function funder the modulus is 1/?n and it could not be improved except for some subclasses of functions. Contrary to this claim, in this paper, we will show that a better order of approximation can be obtained with the help of modulus of continuity.

Using maximum instead of sum, nonlinear Baskakov operator of maximum product kind is introduced by Bede et al. The present paper deals with the approximation processes for this operator. Especially in , it was indicated that the order of approximation of this operator to the function f under the modulus is and it could not be improved except for some subclasses of functions. Contrary to this claim under some circumstances, we will show that a better order of approximation can be obtained with the help of classical and weighted modulus of continuities.

In the present paper, some direct and inverse theorems relating to a generalization of positive linear operators are given. Also some rates of weighted statistical convergence are computed by means of a weighted modulus of continiuty.

In this paper, we investigate a variant of the Jain operators, which preserve the linear functions. We compute the rate of convergence of these operators with the help of K-functional. We also introduce modifications of the Jain operators based on the models in [4] and [10]. These modified operators yield better error estimates than the Jain operators.

We introduce a Stancu type generalization of the Lupaş operators based on the q-integers, rate of convergence of this modification are obtained by means of the modulus of continuity, Lipschitz class functions and Peetre’s K-functional. We will also introduce r-th order generalization of these operators and obtain its statistical approximation properties.

In the present paper, we introduce a new sequence of linear positive operators with the help of generating functions. We obtain some Korovkin type approximation properties for these operators and compute rates of convergence by means of the first and second order modulus of continuities and Peetre´s K-functional. In order to obtain explicit expressions for the first and second moment of our operators, we obtain a functional differential equation including our operators. Furthermore, we deal with a modification of our operators converging to integral of function f on the interval (0,1).