On q-Bleimann , Butzer , and Hahn-Type Operators

and Applied Analysis 3 In this study, we consider a generalization of q-Bleimann, Butzer, andHahnoperators in the sense of [26], we investigate uniform convergence of {Lτ n,q (f; x)} n∈N to f(x) on [0,∞) for f ∈ Hτ ω , and we obtain the degree of approximation. Moreover, we study shape preserving properties under τconvexity of the function. Our results show that the new operators are sensitive to the rate of convergence to f, depending on the selection of τ. For the particular case τ(x) = x, the previous results for q-Bleimann Butzer and Hahn operators are obtained. In order to ensure that the convergence properties holds, the author will assume q = q n is a sequence such that q n → 1 as n → ∞ for 0 < q n < 1, as in [22]. Definition 1. Let x 0 , x 1 , . . . , x n be distinct points in the domain of f. Denote f [x 0 , x 1 , . . . , x n ] = n

The -integer [] and the -factorial []! are defined by [0] = 0, and respectively, where  > 0. For integers  ≥  ≥ 0 the binomial coefficient is defined as Moreover, Euler identity is given by Aral and Dogru [22] constructed the -Bleimann, Butzer, and Hahn operators as where and  is defined on the semiaxis [0, ∞).The authors studied Korovkin-type approximation properties by using the test functions (/(1 + )) V for V = 0, 1, 2.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 as where   is the th Bernstein polynomial,  ∈ [0, 1],  ∈ [0, 1], and  is a function that is continuously differentiable of infinite order on [0, 1] such that (0) = 0, (1) = 1, and   () > 0 for  ∈ [0, 1].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 Korovkintype 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   [0, ∞) [27].
Let  be a general modulus of continuity, satisfying the following properties: (a)  is continuous, nonnegative, and increasing function on [0, ∞), The space of all real valued functions  defined on [0, ∞) satisfying (10) for all ,  ∈ [0, ∞) is denoted by    .It is clear from condition (b) that we have and one can get from the condition (a) that for any  > 0 where [||] denotes the greatest integer that is not greater than .Now we define a new generalization of -Bleimann, Butzer, and Hahn operators for  ∈ [0, ∞) by where and  is a continuously differentiable function on [0, ∞) 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  =  1 , then   , =  , .Obviously, we have Abstract and Applied Analysis 3 In this study, we consider a generalization of -Bleimann, Butzer, and Hahn operators in the sense of [26], we investigate uniform convergence of {  , (; )} ∈N to () on [0, ∞) 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   → 1 as  → ∞ for 0 <   < 1, as in [22].
where  remains fixed and  takes all values from 0 to , excluding .
In [25] Cárdenas-Morales et al. introduced the following definition of -convexity of a continuous function.

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.
Now we are ready to give the following theorem.

Shape Preserving Properties
Theorem 7. Let  be a -convex function that is nonincreasing on [0, ∞); then one has for  ∈ N.
Proof.From (17), we have Using the equality