On q-analogue of a complex summation-integral type operators in compact disks

Recently, Gupta and Wang introduced certain q-Durrmeyer type operators of real variable x ∈ [0, 1] and studied some approximation results in the case of real variables. Here we extend this study to the complex variable for analytic functions in compact disks. We establish the quantitative Voronovskaja type estimate. In this way, we put in evidence the over convergence phenomenon for these q-Durrmeyer polynomials; namely, the extensions of approximation properties (with quantitative estimates) from the real interval [0,1] to compact disks in the complex plane. Some of these results for q = 1 were recently established in Gupta-Yadav.

Mathematical subject classification (2000): 30E10; 41A25.

In the recent years applications of q-calculus in the area of approximation theory and number theory is an active area of research. Several researchers have proposed the q analogue of exponential, Kantorovich and Durrmeyer type operators. Also Kim [1,2] used q-calculus in area of number theory. Recently, Gupta and Wang [3] proposed certain q-Durrmeyer operators in the case of real variables. The aim of the present article is to extend approximation results for such q-Durrmeyer operators to the complex case. The main contributions for the complex operators are due to Gal; in fact, several important results have been complied in his recent monograph [4]. Also very recently, Gal and Gupta [5-7] have studied some other complex Durrmeyer type operators, which are different from the operators considered in the present article.

We begin with some notations and definitions of q-calculus: For each nonnegative integer k, the q-integer [k]_q and the q-factorial [k]_q! are defined by

and

respectively. For the integers n, k, n ≥ k ≥ 0, the q-binomial coefficients are defined by

In this article, we shall study approximation results for the complex q-Durrmeyer operators (introduced and studied in the case of real variable by Gupta-Wang [3]), defined by

where , n = 1, 2, . . . ; q ∈ (0, 1) and , q-Bernstein basis functions are defined as

also in the above q-Beta functions [8] are given as

Throughout the present article we use the notation and by H(D_R), we mean the set of all analytic functions on with for all z ∈ D_R. The norm ||f||_r = max{|f(z)| : |z| ≤ r}. We denote π_p,n (q; z) = M_n,q (e_p; z) for all e_p = t^p, ∪ {0}.

In what follows, we shall study the approximation properties of the operators M_n,q(f; z), which is extension of the results studied in [10]. Further, for these operators we will estimate an upper bound, a quantitative Voronovskaja-type asymptotic formula, and exact order of approximation on compact disks.

To prove the results of following sections, we need the following basic results.

Lemma 1. Let q ∈ (0, 1). Then, π_m,n(q; z) is a polynomial of degree ≤ min (m, n), and

where c_s(m) ≥ 0 are constants depending on m and q, and B_n,q(f; z) is the q-Bernstein polynomials given by .

Proof. By definition of q-Beta function, with , we have

For m =1, we find

thus the result is true for m = 1 with c₁(1) = 1 > 0.

Next for m = 2, with [k + 1]_q = 1 + q[k]_q, we get

thus the result is true for m = 2 with c₁(2) = 1 > 0, c₂(2) = q > 0.

Similarly for m = 3, using [k + 2]_q = [2]_q + q²[k]_q and [k + 1]_q = 1 + q[k]_q we have

where c₁(3) = [2]_q > 0, c₂(3) = 2q² + q > 0, and c₃(3) = q³ > 0.

Continuing in this way the result follows immediately for all m ∈ N. □

Lemma 2. Let q ∈ (0, 1). Then, for all m, , we have the inequality

Proof. By Lemma 1, with e_m = t^m, we have

Also

It immediately follows that p_n,k(q; 1) = 0, k = 0, 1, 2, . . . , n - 1 and p_n,n(q; 1) = 1. Thus, we obtain

□

Corollary 1. Let r ≥ 1 and q ∈ (0, 1). Then, for all m, and |z| ≤ r we have |π_m,n(q; z)| ≤ r^m.

Proof. By using the methods Gal [4], p. 61, proof of Theorem 1.5.6, we have |B_n,q (e_s; z)| ≤ r^s, By Lemma 2 and for all and |z| ≤ r,

□

Lemma 3. Let q ∈ (0, 1) then for , we have the following recurrence relation:

Proof. By simple computation, we have

and

Using these identities, it follows that

Let us denote . Then, the last q-integral becomes

and hence

Therefore,

Finally, using the identity [p + 2]_q + [n]_q q^p+2 = [n + p + 2]_q, we get the required recurrence relation. □

If P_m (z) is a polynomial of degree m, then by the Bernstein inequality and the complex mean value theorem, we have

The following theorem gives the upper bound for the operators (1.1).

Theorem 1. Let for all |z| < R and let 1 ≤ r ≤ R, then for all |z| ≤ r, q ∈ (0, 1) and ,

where .

Proof First, we shall show that . If we denote with , then by the linearity of M_n,q, we have

Thus, it suffice to show that for any fixed and |z| ≤ r with r ≥ 1, lim_m→∞ M_n,q(f_m, z) = M_n,q(f; z). But this is immediate from lim_m→∞ ||f_m - f||_r = 0 and by the inequality

where

Since, π_0,n(q; z) = 1, we have

Now using Lemma 3, for all p ≥ 1, we find

However

Combining the above relations and inequalities, we find

From the last inequality, inductively it follows that

Thus, we obtain

which proves the theorem. □

Remark 1. Let q ∈ (0, 1) be fixed. Since, Theorem 1, is not a convergence result. To obtain the convergence one can choose 0 < q_n < 1 with q_n ↗ 1 as n → ∞. In that case as n → ∞(see Videnskii[9], formula(2.7)), from Theorem 1 we get , uniformly for |z| ≤ r, and for any 1 ≤ r < R.

Here we shall present the following quantitative Voronovskaja-type asymptotic result:

Theorem 2. Suppose that f ∈ H(D_R), R > 1. Then, for any fixed r ∈ [1, R] and for all , |z| ≤ r and q ∈ (0, 1), we have

where , and

Proof. In view of the proof of Theorem 1, we can write . Thus

for all z ∈ D_R, . If we denote

then E_k,n(q; z) is a polynomial of degree ≤ k, and by simple calculation and using Lemma 3, we have

where

Obviously as 0 < q < 1, it follows that

Next with [n + k + 1]_q = [k - 1]_q + q^k-1[n]_q + q^n+k-1 + q^n+k, we have

and

Thus,

Now we will estimate X_3,q,n(k):

Hence, it follows that

Thus,

for all k ≥ 1, and |z| ≤ r.

Next, using the estimate in the proof of Theorem 1, we have

for all k, , | z | ≤ r, with 1 ≤ r.

Hence, for all k, , k ≥ 1 and | z | ≤ r, we have

However, since and , it follows that

Now we shall compute an estimate for , k ≥ 1. For this, taking into account the fact that E_k-1,n (q; z) is a polynomial of degree ≤ k - 1, we have

Thus,

and

where

for all | z | ≤ r, k ≥ 1, , where

Hence, for all | z | ≤ r, k ≥ 1,

where B_k,r is a polynomial of degree 3 in k defined as

But E_0,n (q; z) = 0, for any z ∈ C, and therefore by writing the last inequality for k = 1, 2, . . . we easily obtain step by step the following

Therefore, we can conclude that

As and the series is absolutely convergent in |z| ≤ r, it easily follows that , which implies that This completes the proof of theorem. □

Remark 2. For q ∈ (0, 1) fixed, we have as n → ∞, thus Theorem 2 does not provide convergence. But this can be improved by choosing with q_n ↗ 1 as n → ∞. Indeed, since in this case as n → ∞ and from Theorem 2, we get

Our next main result is the exact order of approximation for the operator (1.1).

Theorem 3. Let , , R > 1, and let f ∈ H(D_R), R > 1. If f is not a polynomial of degree 0, then for any r ∈ [1, R), we have

where the constant C_r(f) > 0 depends on f, r and on the sequence , but it is inde-pendent of n.

Proof. For all and , we have

We use the following property.

to obtain

By the hypothesis, f is not a polynomial of degree 0 in D_R, we get ||e₁(1-e₁)f"-2e₁ f'||_r > 0. Supposing the contrary, it follows that z(1 - z)f"(z) - 2zf'(z) = 0 for all |z| ≤ r, that is (1 - z)f"(z) - 2f'(z) = 0 for all | z| ≤ r with z ≠ 0. The last equality is equivalent to [(1 - z) f'(z)]' - f'(z) = 0, for all |z| ≤ r with z ≠ 0. Therefore, (1 - z) f'(z) - f(z) = C, where C is a constant, that is, , for all |z| ≤ r with z ≠ 0. But since f is analytic in and r ≥ 1, we necessarily have C = 0, a contradiction to the hypothesis.

But by Remark 2, we have

with as n → ∞. Therefore, it follows that there exists an index n₀ depending only on f, r and on the sequence (q_n)_n, such that for all n ≥ n₀, we have

which implies that

For 1 ≤ n ≤ n₀ - 1, we clearly have

where , which finally implies

where

□

Combining Theorem 3 with Theorem 1 we get the following.

Corollary 2. Let for all , R > 1, and suppose that f ∈ H(D_R). If f is not a polynomial of degree 0, then for any r ∈ [1, R), we have

where the constants in the above equivalence depend on f, r, (q_n)_n, but are independent of n.

The proof follows along the lines of [7].

Remark 3. For 0 ≤ α ≤ β, we can define the Stancu type generalization of the operators (1.1) as

The analogous results can be obtained for such operators. As analysis is different, it may be considered elsewhere.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

The authors are thankful to the referees for valuable suggestions leading to overall improvements in article.

1. Kim, T: Some Identities on the q-integral representation of the product of several q-Bernstein-type polynomials, abstract and applied analysis. Article ID 634675, 11 (2011). doi:10.1155/2011/634675 (2011)

2. Kim, T: q-generalized Euler numbers and polynomials. Russ J Math Phys. 13(3), 293–298 (2006). Publisher Full Text

3. Gupta, V, Wang, H: The rate of convergence of q-Durrmeyer operators for 0 < q <1. Math Meth Appl Sci. 31, 1946–1955 (2008). Publisher Full Text

4. Gal, SG: Approximation by Complex Bernstein and Convolution-Type Operators. World Scientific Publ Co, Singapore (2009)

5. Gal, SG, Gupta, V: Approximation by a Durrmeyer-type operator in compact disk. Annali Dell Universita di Ferrara. 57, 261–274 (2011). Publisher Full Text

6. Gal, SG, Gupta, V: Quantative estimates for a new complex Durrmeyer operator in compact disks. Appl Math Comput. 218(6), 2944–2951 (2011)

7. Gal, S, Gupta, V, Mahmudov, NI: Approximation by a complex q Durrmeyer type operators. Ann Univ Ferrara. 58(1), 65–82 (2012). Publisher Full Text

8. Kac, V, Cheung, P: Quantum Calculus, Springer, New York (2002)

9. Videnskii, VS: On q-Bernstein polynomials and related positive linear operators (in Russian). Problems of Modern Mathematics and Mathematical Education, St.-Petersburg. 118–126 (2004)

10. Gupta, V, Yadav, R: Approximation by a complex summation-integral type operator in compact disks. In: Mathematica Slovaca (to appear)