The well known Meyer-König and Zeller operators are defined for functions by  - 
The Meyer-König and Zeller operators  - , the Durrmeyer-type  -  have been the object of several investigations in approximation theory. The estimation of moments, the direct and inverse approximation properties were studied. Recently, many new modified types  -  have been constructed for different function spaces. Gal, Mahmudov, Opris etc.     obtained the quantitative approximation estimates by complex Bernstein-type, Szász-type operators in compact disks.
The goal of this paper is to extend the results to complex Meyer-König and Zeller operators defined as follows: For analytic functions ,
We will obtain the following estimates for the complex Meyer-König and Zeller operators.
Theorem 1. Suppose that is analytic in and continuous in , that is, , for all . Let , for all and , we have
Theorem 2. Under the conditions of Theorem 1, for all and , we have the following Voronovskaja type results
1) , for ;
2) , for .
Theorem 3. Under the hypothesis of Theorem 2, if f is not a polynomial of degree and the series , then for , we have
The paper is organized as the following: In Section 2, we are going to promote the relationship between the Meyer-König and Zeller and Baskakov operators to complex space. In Section 3, we will study the approximation by the complex Baskakov operators. In Section 4, we will give the proof of Theorems 1 - 3. In Section 5, we will give the conclusion of this paper.
2. The Connection between the Complex Meyer-König and Zeller and Baskakov Operators
The proof is based on the connection between Meyer-König and Zeller and Baskakov operators. V. Totik was the first to use it  in the study of Meyer-König and Zeller operators, and many other afterward, see e.g.     . In this section, the connection will be extended to complex space. We will study a transformation mapping functions defined on into functions defined on . The operator will allow us to relate the results for the complex Baskakov operators to their counterparts for the complex Meyer-König and Zeller operators. We will consider variables and functions defined on as respectively, and their analogs defined on , as the later will be denoted with . We consider the weight functions
defined for real values of the parameters . We will utilize the change given by
Remark 1. , where , . For example: .
Then, its inverse change is
Remark 2. From the definition of and , we have that the change and are linear fractional transformations and conformal mappings.
A function g defined on is transformed to a function f defined on by
The inverse operator transforming a function f defined on to a function g defined on is
When a product of two functions is treated, that means, the associated operator is defined by
and its inverse is defined by
For , , we have
The operators and have the following properties. From the definition (1)-(3), we yield immediately.
Proposition 1. Let , denote the spaces of all functions defined on and respectively. Then and are linear operators.
Proposition 2. Let be a weight in , ,
Then the mapping is a linear correspondence with ,
Proof. From the definition of the mapping (2) and the operator (4), combining the Proposition 1, we get the mapping is a linear correspondence.
Noting that the relation
one can get the desired result.
The following proposition is very important, it gives the connection between the complex Meyer-König and Zeller operators and the complex Baskakov operators
Proposition 3. For every f such that one of the series in (6) is convergent, for every , we have
Proof. From the definition of the operator , Proposition 1 and the identities
valid for , we have (6).
Proposition 4. Under the conditions of Proposition 3, we have
Proof. From Proposition 3, relations ((4), (1), (3)), we obtain for and ,
Remark 3. If the weight (i.e. ), the corresponding weight to is .
Then, we have the following auxiliary results.
Lemma 2.1. Under the conditions of Proposition 3, , , we have
Lemma 2.2.  Denoting and , is a polynomial of degree p, , we have the recurrence formula
3. Weighted Approximation by the Complex Baskakov Operators
Theorems 1 - 3 will be proved in Section 4 by transferring the corresponding results for the complex Baskakov operators. In this section, we will prove some properties of the complex Baskakov operators. The first main result of this section is the following theorem for upper bound.
Theorem 3.1. Suppose that is continuous in and analytic in , i.e. . Let , for all , we have
Proof. By using the recurrence relation of Lemma 2.2, for all , we have
From this we immediately get the recurrence formula
To estimate , we wil use the relation  p. 7:
for all , where is a polynomial of degree . Then, we get
We will prove the following relation by mathematical induction with respect to p:
Indeed for , . Suppose that it is true for , that is,
Now for , by the relations ((7), (8)), we have
It remains to prove that for
By mathematical induction that the last inequality holds true for all and . From the hypothesis on g, it follows that is analytic in , we write
Theorem 3.2. Under the conditions of Theorem 3.1, let , for all , we have the following Voronovskaja type formula
1) for , ;
2) for , .
Proof. Case I. For , noting that and and , we have
Using the recurrence relation of Lemma 2.2, we write
Noting that , for , we have
By simple computation, we get
Thus, for all , , , we have
Using the estimate in the proof of Theorem 3.1, for all and , we have
Now we shall estimate for. Noting that is a polynomial of degree, combining the Bernstein's inequality, we have
we obtain step by step following
which follows that
Case 2. For, in the proof of Case 1, the relation (9) should be changed to
and the relation (10) should be changed to
4. The Proof of Theorems 1 - 3
The Proof of Theorem 1. Combining Lemma 2.1 and Theorem 3.1, we can obtain Theorem 1.
The Proof of Theorem 2. From Lemma 2.1 and Theorem 3.2, we have Theorem 2.
In what follows we obtain the exact degree in the approximation by.
Theorem 4.1. Suppose that the hypothesis on the function f and Theorem 2. If f is not a polynomial of degree and the series, then holds, where depends only on f and r.
Proof. For all, we can write
Applying the inequality, we obtain
Since f is not a polynomial of degree in, we get. Indeed, supposing the contrary, it follows that for all, which implies for all. Since f is analytic in, this means that for all, that is f is a polynomial of degree, a contradiction with the hypothesis.
Now by Theorem 2, for, we have
Choose, such that for all, we have
which implies for all,
For, we have
The Proof of Theorem 3. From Lemma 2.1, Theorem 4.1 and Theorem 1, we can obtain Theorem 3.
In this paper, the properties of approximation are studied by using the general relation between the Meyer-König and Zeller and Baskakov operators. The geometric properties (the shap-preserving) of such complex operators still remain to be studied.
We thank the Editor and the referee for their comments. The work is partially supported by NSF of China (11571089, 11871191) and NSF of Hebei Province (2012205028; ZD2019053). The project supported by science foundation of Hebei Normal University.
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