Abstract
This paper deals with approximating properties and convergence results of the complex q-Balázs-Szabados operators attached to analytic functions on compact disks. The order of convergence and the Voronovskaja-type theorem with quantitative estimate of these operators and the exact degree of their approximation are given. Our study extends the approximation properties of the complex q-Balázs-Szabados operators from real intervals to compact disks in the complex plane with quantitative estimate.
MSC:30E10, 41A25.
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1 Introduction
In the recent years, applications of q-calculus in the area of approximation theory and number theory have been an active area of research. Details on q-calculus can be found in [1–3]. Several researchers have purposed the q-analogue of Stancu, Kantorovich and Durrmeyer type operators. Gal [4] studied some approximation properties of the complex q-Bernstein polynomials attached to analytic functions on compact disks.
Also very recently, some authors [5–7] have studied the approximation properties of some complex operators on complex disks. Balázs [8] defined the Bernstein-type rational functions and gave some convergence theorems for them. In [9], Balázs and Szabados obtained an estimate that had several advantages with respect to that given in [8]. These estimates were obtained by the usual modulus of continuity. The q-form of these operator was given by Doğru. He investigated statistical approximation properties of q-Balázs-Szabados operators [10].
The rational complex Balázs-Szabados operators were defined by Gal [4] as follows:
where with , is a function, , , , , and .
He obtained the uniform convergence of to on compact disks and proved the upper estimate in approximation of these operators. Also, he obtained the Voronovskaja-type result and the exact degree of its approximation.
The goal of this paper is to obtain convergence results for the complex q-Balázs-Szabados operators given by
where is uniformly continuous and bounded on , , , , , , and for .
These operators are obtained simply replacing x by z in the real form of the q-Balázs-Szabados operators introduced in Doğru [10].
The complex q-Balázs-Szabados operators are well defined, linear, and these operators are analytic for all and since .
In this paper, we obtain the following results:
-
the order of convergence for the operators ,
-
the Voronovskaja-type theorem with quantitative estimate,
-
the exact degree of the approximation for the operators .
Throughout the paper, we denote with the norm of f in the space of continuous functions on and with the norm of f in the space of bounded functions on .
Also, the many results in this study are obtained under the condition that is analytic in for , which assures the representation for all .
2 Convergence results
The following lemmas will help in the proof of convergence results.
Lemma 1 Let , and . Let us define for all , where . If is uniformly continuous, bounded on and analytic in , then we have the form
for all .
Proof For any , we define
From the hypothesis on f, it is clear that each is bounded on , that is, there exist with , which implies that
that is all with , , are well defined for all .
Defining
it is clear that each is bounded on and that .
From the linearity of , we have
It suffices to prove that
for any fixed , and .
We have the following inequality for all :
where .
Using (1), and , the proof of the lemma is finished. □
Lemma 2 If we denote , then the following formula holds:
where β is a fixed real number and .
Proof We can write as follows:
In [3] (see p.10, Proposition 3.3), we already have the following formula:
Using (2) and (3), we get
From (4), we obtain the result. □
Lemma 3 We have the following recurrence formula for the complex q-Balázs-Szabados operators :
where for all , and .
Proof Firstly, we calculate as follows:
Considering Lemma 2 and using in (5), we get
From (6), the proof of the lemma is finished. □
Corollary 1 ([11], p.143, Corollary 1.10.4)
Let , where is a polynomial of degree ≤k, and we suppose that for all . If , then for all we have
Under hypothesis of the corollary above, by the mean value theorem [12] in complex analysis, we have
Lemma 4 Let , and . For all , and , we have
Proof Taking the absolute value of the recurrence formula in Lemma 3 and using the triangle inequality, we get
In order to get an upper estimate for , by using (7), we obtain
Under the condition , it holds , which implies
Applying (9) to (8) and passing to norm, we get
From the hypothesis of the lemma, we have , , and , which implies
Taking step by step , we obtain
Using and replacing with k, the proof of the lemma is finished. □
Let be a sequence satisfying the following conditions:
Now we are in a position to prove the following convergence result.
Theorem 1 Let be a sequence satisfying the conditions (10) with for all , and let , and . If is uniformly continuous, bounded on and analytic in and there exist , with (which implies for all ), then the sequence is uniformly convergent to f in .
Proof From Lemma 2 and Lemma 6, for all and , we have
where the series is convergent for .
Since for all (see [10]), by Vitali’s theorem (see [13], p.112, Theorem 3.2.10), it follows that uniformly converges to in . □
We can give the following upper estimate in the approximation of .
Theorem 2 Let be a sequence satisfying the conditions (10) with for all , and let , and . If is uniformly continuous, bounded on and analytic in and there exist , with (which implies for all ), then the following upper estimate holds:
where and .
Proof Using the recurrence formula in Lemma 4, we have
For , we get
Using (9), , and , we obtain
Since for all , we can write
Taking step by step, finally we arrive at
which implies
Choosing , we obtain the desired result.
Here the series is convergent for and the series is absolutely convergent in , it easily follows that . □
The following lemmas will help in the proof of the next theorem.
Lemma 5 For all , we have
where for .
Proof (12) and (13) are obtained simply replacing x by z in Lemma 3.1 and Lemma 3.2 in [10]. Also, using and and replacing x by z in Lemma 3.3 in [10], (14) is obtained. □
Lemma 6 For all , the following equalities for the operators hold:
where for .
Proof From Lemma 5, the proof can be easily got, so we omit it. □
Now, we present a quantitative Voronovskaja-type formula.
Let us define
Theorem 3 Let be a sequence satisfying the conditions (10) with for all , , and . If is uniformly continuous, bounded on and analytic in and there exist , with (which implies for all ), then for all and , we have
where and is a fixed real number.
Proof From Lemma 1 and the analyticity of f, we can write
where
Using Lemma 5, we easily obtain that .
Combining (19) with the recurrence formula in Lemma 3, a simple calculation leads us to the following recurrence formula:
where
In the following results, will denote fixed real numbers for .
Under the hypothesis of Theorem 3, we have
Using (21)-(24), for , we get
On the other hand, for , we have
Taking into account (11) in the proof of Theorem 2, we obtain
Considering (25) and (26) in (20), we get
Since , taking in the last inequality step by step, finally we arrive at
Finally, considering (27) in (18) and using , the proof of the theorem is complete. □
Remark 1 For , since as , therefore and as . If a sequence satisfies the conditions (10), then as ; therefore and as .
Under the conditions (10), Theorem 2 and Theorem 3 show that uniformly converges to in .
From Theorem 2 and Theorem 3, we get the following consequence.
Theorem 4 Let be a sequence satisfying the conditions (10) with for all , , , and . Suppose that is uniformly continuous, bounded on and analytic in and there exist , with (which implies for all ). If f is not a polynomial of degree ≤1, then for all we have
Proof We can write
where
and
and also is a sequence of analytic functions uniformly convergent to zero for all .
Since as , and taking into account Theorem 3, it remains only to show that for sufficiently large n and for all , we have , where ρ is independent of n.
If , then the term as , while the other terms converge to zero, so there exists a natural number with so that for all and , we have
If , then the term as , while the other terms converge to zero. So, there exists a natural number with so that for all and , we have
In the case of , that is, , we obtain, as , so that the case remains unsettled.
Choosing , considering (31) and (32), for all , we get
For all , we get
with , which finally implies
for all , with .
From (33) and Theorem 3, the proof is complete. □
Remark 2 Recently, it is much more interesting to study these operators in the case . Authors continue to study that case.
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The main idea of this paper is proposed by NI. All authors contributed equally in writing this article and read and approved the final manuscript.
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İspir, N., Yıldız Özkan, E. Approximation properties of complex q-Balázs-Szabados operators in compact disks. J Inequal Appl 2013, 361 (2013). https://doi.org/10.1186/1029-242X-2013-361
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DOI: https://doi.org/10.1186/1029-242X-2013-361