In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called $(\alpha,q)$ -Bernstein operators, denoted by $T_{n,q,\alpha}(f)$ . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of $T_{n,q,\alpha}(f)$ to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to $f(x)$ . We also obtain the monotonicity with n and q of $T_{n,q,\alpha}(f)$ .

In this note, we give an elaboration of a basic problem on convergence theorem of ( p , q ) $(p, q)$ -analogue of Bernstein-type operators. By some classical analysis techniques, we derive an exact class of ( p n , q n ) $(p_{n},q_{n})$ -integer satisfying lim n → ∞ [ n ] p n , q n = ∞ $\lim _{n\to\infty }[n]_{p_{n},q_{n}}=\infty$ with lim n → ∞ p n = 1 $\lim _{n\to\infty}p_{n}=1$ and lim n → ∞ q n = 1 $\lim _{n\to\infty}q_{n}=1$ under 0 < q n < p n ≤ 1 $0< q_{n}< p_{n}\leq1$ . Our results provide an erratum to corresponding results on ( p , q ) $(p,q)$ -analogue of Bernstein-type operators that appeared in recent literature.

In the present paper, Kantorovich type $\lambda$ -Bernstein operators via (p, q)-calculus are constructed, and the first and second moments and central moments of these operators are estimated in order to achieve our main results. An A-statistical convergence theorem and the rate of A-statistical convergence theorems are obtained according to some analysis methods and the definitions of A-statistical convergence, the rate of A-statistical convergence and modulus of smoothness.