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Approximation by Genuine -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case
This paper deals with approximating properties of the newly defined -generalization of the genuine Bernstein-Durrmeyer polynomials in the case , which are no longer positive linear operators on . Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine -Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials is of order versus for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine -Bernstein-Durrmeyer for . This paper represents an answer to the open problem initiated by Gal in (2013, page 115).
In several recent papers, convergence properties of complex -Bernstein polynomials, proposed by Phillips , attached to an analytic function in closed disks, were intensively studied. Ostrovska [2, 3] and Wang and Wu [4, 5] have investigated convergence properies of in the case . In the case , the -Bernstein polynomials are no longer positive operators; however, for a function analytic in a disc , it was proved in  that the rate of convergence of to has the order (versus for the classical Bernstein polynomials). Moreover, Ostrovska  obtained Voronovskaya-type theorem for monomials. If , then qualitative Voronovskaja-type theorem and saturation results for complex -Bernstein polynomials were obtained by Wang and Wu . Wu  studied saturation of convergence on the interval for the -Bernstein polynomials of a continuous function for arbitrary fixed .
Genuine Bernstein-Durrmeyer operators were first considered by Chen  and Goodman and Sharma  around 1987. In recent years, the genuine Bernstein-Durrmeyer operators have been investigated intensively by a number of authors. Among the many papers written on the genuine Bernstein-Durrmeyer operators, we mention here only the ones by Gonska et al. , Parvanov and Popov , Sauer , Waldron , and the book of Păltănea .
On the other hand, Gal  obtained quantitative estimates of the convergence and of the Voronovskaja-type theorem in compact disks, for the complex genuine Bernstein-Durrmeyer polynomials attached to analytic functions. Besides, in other very recent papers, similar studies were done for complex Bernstein-Durrmeyer operators in Anastassiou and Gal , for complex Bernstein-Durrmeyer operators based on Jacobi weights in Gal , for complex genuine -Bernstein-Durrmeyer operators () by Mahmudov , and for other kinds of complex Durrmeyer operators in Mahmudov  and Gal et al. . It should be stressed out that study of -Durrmeyer-type operators () in the real case was first initiated by Derriennic .
Also, for the case , exact quantitative estimates and quantitative Voronovskaja-type results for complex -Lorentz polynomials, -Stancu polynomials , -Stancu-Faber polynomials, -Bernstein-Faber polynomials, -Kantorovich polynomials , -Szász-Mirakjan operators  obtained by different researchers are collected in the recent book of Gal . In this book the definition and study of complex -Durrmeyer-kind operators for presented an open problem. This paper presents a positive solution to this problem.
In this paper we define the genuine -Bernstein-Durrmeyer polynomials for . Note that similar to the -Bernstein operators the genuine -Bernstein-Durrmeyer operators in the case are not positive operators on . The lack of positivity makes the investigation of convergence in the case essentially more difficult than that for . We present upper estimates in approximation and we prove the Voronovskaja-type convergence theorem in compact disks in , centered at origin, with quantitative estimate of this convergence. These results allow us to obtain the exact degrees of approximation by complex genuine -Bernstein-Durrmeyer polynomials. Our results show that approximation properties of the complex genuine -Bernstein-Durrmeyer polynomials are better than approximation properties of the complex Bernstein-Durrmeyer polynomials considered in .
2. Main Results
We begin with some notations and definitions of -calculus; see, for example, [24, 25]. Let . For any , the -integer is defined by and the -factorial is defined by For integers , the -binomial is defined by For we obviously get , , and . Moreover
For fixed , , we denote the -derivative of by
The -analogue of integration in the interval (see ) is defined by Let be a disc in the complex plane . Denote by the space of all analytic functions on . For we assume that for all . The norm . We denote for all .
Definition 1. For , the genuine -Bernstein-Durrmeyer operator is defined as follows: where for the sum is empty; that is, it is equal to .
are linear operators reproducing linear functions and interpolating every function at and . The genuine -Bernstein-Durrmeyer operators are positive operators on for , and they are not positive for . As a consequence, the cases and are not similar to each other regarding the convergence. For and we recapture the classical () genuine Bernstein-Durrmeyer polynomials.
We start with the following quantitative estimates of the convergence for complex -Bernstein-Durrmeyer polynomials attached to an analytic function in a disk of radius and center .
Theorem 2. Let , , and . Then for all one has
Theorem 2 says that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials () is of order versus for the classical genuine Bernstein-Durrmeyer polynomials; see .
The Voronovskaja theorem for the real case with a quantitative estimate is obtained by Gonska et al.  in the following form: and, for all , . For the complex genuine -Bernstein-Durrmeyer () a quantitative estimate is obtained by Gal  () and Mahmudov  () in the following form: and, for all , .
To formulate and prove the Voronovskaja-type theorem with a quantitative estimate in the case we introduce a function .
Let and let . For , we define And, for ,
The next theorem gives Voronovskaja-type result in compact disks; for complex -Bernstein-Durrmeyer polynomials attached to an analytic function in , and center in terms of the function .
Theorem 3. Let , , and . The following Voronovskaja-type result holds: For all , .
Theorem 4. Let , , and . If is not a polynomial of degree ≤1, the estimate, holds, where the constant depends on , , and but is independent of .
From Theorem 3 we conclude that, for , in and therefore . Furthermore, we have the following saturation of convergence for the genuine -Bernstein-Durrmeyer polynomials for fixed .
Theorem 5. Let , . If a function is analytic in the disc , then for infinite number of points having an accumulation point on if and only if is linear.
The next theorem shows that , is continuous in the parameter for , .
Theorem 6. Let and . Then, for any , uniformly on .
3. Auxiliary Results
The -analogue of beta function for (see ) is defined as Since we consider the case , we need to use as follows: Also, it is known that For , we have Thus, we get the following formula for : Note that, for , we have
Lemma 7. is a polynomial of degree less than or equal to and
Proof. From (20) it follows that Now using where , , are the constants independent of , we get Since is a polynomial of degree less than or equal to and , , it follows that is a polynomial of degree less than or equal to .
Lemma 8. The numbers , given by (24), enjoy the following properties:
Also, the following lemma holds.
Lemma 9. For all the identity, holds.
Proof. It follows from end points interpolation property of and . Indeed
Lemma 9 implies that for all and we have
For our purpose first we need a recurrence formula for .
Lemma 10. For all and one has
Proof. By simple calculation we obtain (see ) It follows that which implies the recurrence in the statement.
Let Using the recurrence formula (30) we prove two more recurrence formulas.
Lemma 11. For all and one has where
Proof. From the recurrence formula in Lemma 10, for all , we get where Again by simple calculation we obtain where and can be simplified as follows:
Lemma 12. Let and . The function has the following representation:
Proof. Using the following identity: we get where .
4. Proofs of the Main Results
Firstly we prove that . Indeed denoting with , by the linearity of , we have and it is sufficient to show that, for any fixed and with , we have . But this is immediate from , the norm being defined as , and from the inequality valid for all , where Therefore we get as and .
Proof of Theorem 2. From the recurrence formula (34) and the inequality (29) for we get It is known that, by a linear transformation, the Bernstein inequality in the closed unit disk becomes which, combined with the mean value theorem in complex analysis, implies for all , where is a complex polynomial of degree ≤. It follows that By writing the last inequality for , we easily obtain, step by step, the following: It follows that
The second main result of the paper is the Voronovskaja-type theorem with a quantitative estimate for the complex version of genuine -Bernstein-Durrmeyer polynomials.