Abstract
Sequence of -Bleimann, Butzer, and Hahn operators which is based on a continuously differentiable function on , with , , has been considered. Uniform approximation by such a sequence has been studied and degree of approximation by the operators has been obtained. Moreover, shape preserving properties of the sequence of operators have been investigated.
1. Introduction
In 1987, Lupaş [1] introduced the first -analogue of Bernstein operators and investigated its approximation and shape preserving properties. Another -generalization of the classical Bernstein polynomials is due to Phillips [2]. After that many generalizations of positive linear operators based on -integers were introduced and studied by several authors. Some are in [3–13].
Bleimann et al. [14] proposed a sequence of positive linear operators defined byfor , where denote the space of all continuous and real valued functions defined on Also the authors proved that as pointwise on for any , where denote the space of all bounded functions from . It is well known that defines a norm on . Moreover, they showed that the convergence is uniform on each compact subset of . In [15], using test functions for , Gadjiev and Çakar stated a Korovkin-type theorem for the uniform approximation of functions belonging to some suitable function spaces by some linear positive operators. As an application of this result, they proved uniform approximation by Bleimann, Butzer, and Hahn operators. Further results concerning such a sequence of operators and its generalizations may be found in [16–19].
Now, we recall some notations from -analysis [20, 21].
The -integer and the -factorial are defined by, andrespectively, where .For integers the -binomial coefficient is defined asMoreover, Euler identity is given by
Aral and Doğru [22] constructed the -Bleimann, Butzer, and Hahn operators aswhereand is defined on the semiaxis The authors studied Korovkin-type approximation properties by using the test functions for . Moreover, they obtained rate of convergence of the operators and proved that rate of the -Bleimann, Butzer, and Hahn operators is better than the classical one. A generalization of the -Bleimann, Butzer, and Hahn operators was introduced by Agratini and Nowak in [23]. In this paper, the authors gave representation of the operators in terms of -differences and investigated some approximation properties.
A Voronovskaja-type result and monotonicity properties of these operators are investigated in [24].
In [25], the authors introduced a new generalization of Bernstein polynomials denoted by and defined aswhere is the th Bernstein polynomial, , , and is a function that is continuously differentiable of infinite order on such that , , and for . Also, the authors studied some shape preserving and convergence properties concerning the generalized Bernstein operators .
In [26], Aral et al. constructed sequences of Szasz-Mirakyan operators which are based on a function . They studied weighted approximation properties and Voronovskaja-type results for these operators. They also showed that the sequence of the generalized Szász-Mirakyan operators is monotonically nonincreasing under the -convexity of the original function. A similar generalization for Bleimann, Butzer, and Hahn operators is studied by Söylemez [27]. Also the class was defined, a Korovkin-type theorem was given for the functions in this class, and uniform convergence of the generalized Bleimann, Butzer, and Hahn operators was obtained [27]. Moreover, the monotonicity properties of the operators were investigated.
Now we recall the definition of that is a subspace of [27].
Let be a general modulus of continuity, satisfying the following properties:(a)is continuous, nonnegative, and increasing function on ,(b),(c)
The space of all real valued functions defined on satisfyingfor all is denoted by
It is clear from condition (b) that we haveand one can get from the condition (a) that for any where denotes the greatest integer that is not greater than .
Now we define a new generalization of -Bleimann, Butzer, and Hahn operators for bywhereand is a continuously differentiable function on such that An example of such a function is given in [26]. Note that, in the setting of the operators (13), we have where the operators are defined by (7). If , then Obviously, we have
In this study, we consider a generalization of -Bleimann, Butzer, and Hahn operators in the sense of [26], we investigate uniform convergence of to on for , and we obtain the degree of approximation. Moreover, we study shape preserving properties under -convexity of the function. Our results show that the new operators are sensitive to the rate of convergence to , depending on the selection of . For the particular case , the previous results for -Bleimann Butzer and Hahn operators are obtained.
In order to ensure that the convergence properties holds, the author will assume is a sequence such that as for , as in [22].
Definition 1. Let be distinct points in the domain of . Denote where remains fixed and takes all values from to , excluding .
Definition 2. A continuous, real valued function is said to be convex in , if for every and for every nonnegative number of such that .
In [25] Cárdenas-Morales et al. introduced the following definition of -convexity of a continuous function.
Definition 3. A continuous, real valued function is said to be -convex in , if is convex in the sense of Definition 2.
2. Approximation Properties
In this section we deal with the promised approximation properties of the sequence of -Bleimann, Butzer, and Hahn operators. In [27], the following Korovkin-type theorem was given.
Theorem 4. Let be a sequence of linear positive operators from to Ifis satisfied for , then for one has
Now we are ready to give the following theorem.
Theorem 5. Suppose that , , and let as If is the operator defined by (13), then for any one has
Proof. According to Theorem 4 we will show that (20) holds for . Obviously, from (17) we easily obtain that Therefore, the conditions (20) are satisfied. By Theorem 4, the proof is completed.
Theorem 6. Let , , and let as Then one hasfor all . Here,
Proof. Firstly, from (12) and (10) we can writeand considering (17), we getEquations (27) and (26) together imply that Using the Cauchy-Schwarz inequality, we obtain By choosing , we get where which concludes the proof.
3. Shape Preserving Properties
Theorem 7. Let be a -convex function that is nonincreasing on ; then one has for .
Proof. From (13), one can writeMoreover, we have the following equalities that are proved in Lemma 3.1 of [24]: which implyFrom and by hypothesis, we have Since is -convex, by using Lemma 3.2 in [24] we obtain for and This proves the theorem.
Theorem 8. Suppose that , and is linear. Then one has
Proof. From (17), we have Using the equality , we now get On the other hand we can writewhich impliesNow taking (42) in (40), we have Thus the proof is completed.
Now, from Theorem 8, we have the following corollary immediately.
Corollary 9. Let be a -convex, nonincreasing function on and is linear. Then one has for any ,
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.