In this paper, we introduce a new type λ-Bernstein operators with parameter $\lambda\in[-1,1]$ , we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of $B_{n,\lambda }(f;x)$ to $f(x)$ , and we see that in some cases the errors are smaller than $B_{n}(f)$ to f.

In this paper, a class of Stancu type Szász-Mirakjan operators are introduced. The approximation properties of the operators are discussed using tools of modulus of continuity, modulus of smoothness and K-functional. The estimation of the Lipschitz function class by the operators is also studied. Later, the Voronvskaya type asymptotic expansion of the operators is established. Finally, we compare the convergence of these newly defined operators for certain functions with certain graphs.

In this paper, the author introduce a class of modified Lupas-Kantorovich type operators which preserve constant and linear functions. By using modulus of continuity, modulus of smooth, K-functional and lipschitz class, the rate of convergence of these operators are derived. Finally, the author present a voronovskaya-type asymptotic formula.

In this paper we introduce two new B?zier variants of the generalized Bernstein type operators. With the help of Ditzian Totik smooth modulus, we discuss a global approximation. Also, we obtain the rate of convergence of the operators by the second order continuous modulus and Peetre K-functional. By using the construction of appropriate functions and the methods of Bojanic-Cheng, the approximation properties of the new defined operators for absolute continuous functions with derivatives equivalent to bounded variation functions is given. Finally, we provide four graphs to illustrate the approximation effect of the newly defined operators.