Báo cáo hóa học: " Research Article Bleimann, Butzer, and Hahn Operators Based on the q-Integers" pdf

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 79410, 12 pages doi:10.1155/2007/79410 Research Article Bleimann, Butzer, and Hahn Operators Based on the q-Integers Ali Aral and Og ¨ un Do ˘ gru Received 29 May 2007; Accepted 9 October 2007 Recommended by Ram N. Mohapatra We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes q- integers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section 3, we show that the rates of convergence of the new operators in uniform norm are better than the classical ones. We also obtain a pointwise estimation in a general Lipschitz-type maximal function space. Finally, we de?fine a generalization of these new operators and study the uniform convergence of them. Copyright © 2007 A. Aral and O. Do ˘ gru. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Recently, in 1997, Phillips [1] used the q-integers in approximation theory where it is con- sidered q-based generalization of classical Bernstein polynomials. It was obtained by re- placing the binomial expansion with the general one, the q-binomial expansion. Phillips has obtained the rate of convergence and Voronovskaja-type asymptotic formulae for these new Bernstein operators based on q-integers. Later, some results are established in due course by Phillips et al. (see [2, 3, 1]). In [4], Barbasu gave Stancu-type generaliza- tion of these operators and II‘inskii and Ostrovska [5] studied their different convergence properties. Also some results on the statistical and ordinary approximation of functions by M eyer-K ¨ onig and Zeller operators based on q-integers can be found in [6, 7], respec- tively. In [8], Bleimann, Butzer, and Hahn introduced the following operators: B n ( f )(x) = 1 (1 +x) n n  k=0 f  k n −k +1   n k  x k , x>0, n ∈ N. (1.1) 2 Journal of Inequalities and Applications There are several studies related to approximation properties of Bleimann, Butzer, and Hahn operators (or, briefly, BBH). There are many approximating operators that their Korovkin-type approximation properties and rates of convergence are investigated. The results involving Korovkin-type approximation properties can be found in [9]withde- tails. In [10], Gadjiev and C¸ akar gave a Korovkin-type theorem using the test function (t/(1 +t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11– 13]. In this paper, we derive a q-integers-type modification of BBH operators that we call q-BBH operators and investigate their Korovkin-type approximation properties by using the test function (t/(1 + t)) ν for ν = 0,1,2. Also, we define a space of generalized Lipschitz- type maximal function and give a pointwise estimation. Then, a Stancu-type formula of the remainder of q-BBH is given. We will also give a generalization of these new oper- ators and study the approximation properties of this generalization. We emphasis that while Bernstein and Meyer-K ¨ onig and Zeller operators based on q-integers depend on a function defined on a bounded interval, these new oper ators are defined on unbounded intervals. Also, these new operators are more flexible than classical BBH operators. That is, depending on the selection of q, rate of convergence of the q-BBH operators is better than the classical one. 2. Construction of the operators We first start by recalling some definitions about q-integers denoted by [ ·]. For any fixed real number q>0 and nonnegative integer r,theq-integer of the number r is defined by [r] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − q r 1 − q , q =1, r, q = 1. (2.1) Also we have [0] = 0. The q-factorial is defined in the following: [r]! = ⎧ ⎪ ⎨ ⎪ ⎩ [r][r − 1]···[1], r = 1,2, , 1, r = 0, (2.2) and q-binomial coefficient is defined as  n r  = [n]! [r]![n − r]! (2.3) for integers n ≥ r ≥ 0. Also, let us recall the following Euler identity (see [14, page 293]): n−1  k=0  1+q k x  = n  k=0 q k(k−1)/2  n k  x k . (2.4) A. Aral and O. Do ˘ gru 3 It is clear that when q = 1, these q-binomial coefficients reduce to ordinary binomial coefficients. According to these explanations, similarly in [6], we define a new Bleimann-, Butzer-, and Hahn-type operators based on q-integers as follows: L n ( f ;x) = 1  n (x) n  k=0 f  [k] [n − k +1]q k  q k(k−1)/2  n k  x k , (2.5) where  n (x) = n−1  s=0  1+q s x  (2.6) and f is defined on semiaxis [0, ∞). Note that taking f ([k]/[n − k + 1]) instead of f ([k]/[n − k +1]q k )in(2.5), we obtain usual generalization of Bleimann, Butzer, and Hahn operators based on q-integers. But in this case, it is impossible to obtain explicit expressions for the monomials t ν and (t/(1 + t)) ν for ν = 1,2. If we define the Bleimann-, Butzer-, and Hahn-type operators as in (2.5), then we can obtain explicit formulas for the monomials (t/(1 + t)) ν for ν = 0,1,2. By a simple calculation, we have q k [n − k +1]= [n +1]− [k], q[k − 1] = [k] − 1. (2.7) From (2.4), (2.5), and (2.7), we have L n (1;x) = 1, (2.8) L n  t 1+t ;x  = 1  n (x) n  k=1 [k] [n +1] q k(k−1)/2  n k  x k = 1  n (x) n  k=1 [n] [n +1] q k(k−1)/2  n − 1 k − 1  x k = [n] [n +1] x 1  n (x) n−1  k=0 q k(k−1)/2  n − 1 k  (qx) k = x x +1 [n] [n +1] . (2.9) We ca n also wr ite L n  t 2 (1 + t) 2 ;x  = 1  n (x) n  k=1 [k] 2 [n +1] 2 q k(k−1)/2  n k  x k = 1  n (x) n  k=2 q[k][k − 1] [n +1] 2 q k(k−1)/2  n k  x k + 1  n (x) n  k=1 [k] [n +1] 2 q k(k−1)/2  n k  x k 4 Journal of Inequalities and Applications = 1  n (x) n−2  k=0 [n][n − 1] [n +1] 2 q k(k−1)/2  n − 2 k   q 2 x  k q 2 x 2 + 1  n (x) n−1  k=0 [n] [n +1] 2 q k(k−1)/2  n − 1 k  (qx) k x = [n][n − 1] [n +1] 2 q 2 x 2 (1 + x)(1 + qx) + [n] [n +1] 2 x x +1 . (2.10) Remark 2.1. Note that if we choose q = 1, then L n operators turn o ut into classical Bleimann, Butzer, and Hahn operators given by (1.1). Also similarly as in [1, 6], to ensure that the convergence properties of L n , we will assume q = q n as a sequence such that q n →1 as n →∞ for 0 <q n < 1. 3. Properties of the operators In this section, we will give the theorems on uniform convergence and rate of convergence of the operators (2.5). As in [10], for this purpose we give a space of function ω of the type of modulus of continuity w hich satisfies the following conditions: (a) ω is a nonnegative increasing function on [0, ∞), (b) ω(δ 1 + δ 2 ) ≤ ω(δ 1 )+ω(δ 2 ), (c) lim δ→0 ω(δ) = 0, and H ω is the subspace of real-valued function and satisfies the following condition. For any x, y ∈ [0,∞),   f (x) − f (y)   ≤ ω      x 1+x − y 1+y      . (3.1) Also H ω ⊂ C B [0,∞), where C B [0,∞) is the space of functions f which is continuous and bounded on [0, ∞) endowed with norm  f  C B = sup x≥0 | f (x)|. It is easy to show that from condition (b), the function ω satisfies the inequality ω(nδ) ≤ nω(δ), n ∈ N, (3.2) and from condition (a) for λ>0, we have ω(λδ) ≤ ω  1+[|λ|]  δ  ≤ (1 + λ)ω(δ), (3.3) where [ |λ|] is the g reatest integer of λ. Remark 3.1. The operator L n maps H ω into C B [0,∞) and it is continuous with respect to supnorm. The properties of linear positive operators acting from H ω to C B [0,∞)andKorovkin- type theorems for them have b een studied by Gadjiev and C¸ akar who have established the following theorem (see [10]). A. Aral and O. Do ˘ gru 5 Theorem 3.2. If A n is the sequence of positive linear operators acting from H ω to C B [0,∞) and sat isfying the following condition for υ = 0,1,2:      A n  t 1+t  υ  (x) −  x 1+x  υ     C B −→ 0, for n −→ ∞ , (3.4) then, for any function f in H ω , one has   A n f − f   C B −→ 0, for n −→ ∞ . (3.5) Theorem 3.3. Let q = q n satisfies 0 <q n < 1 and let q n →1 as n→∞. If L n is defined by (2.5), then for any f ∈ H ω , lim n→∞   L n f − f   C B = 0. (3.6) Proof. Using Theorem 3.2, we see that it is sufficient to verify the follow ing three condi- tions: lim n→∞     L n  t 1+t  υ ;x  −  x 1+x  υ     C B = 0, υ = 0,1,2. (3.7) From (2.8), the first condition of (3.7) is fulfilled for υ = 0. Now it is easy to see that from (2.9),     L n  t 1+t  ;x  − x 1+x     C B ≤     [n] [n +1] − 1     ≤     1 q n − 1 q n [n +1] − 1     , (3.8) and since [n +1] →∞, q n →1asn→∞, condition (3.7)holdsforυ = 1. To verify this condition for υ = 2, consider (2.10). We see that     L n  t 1+t  2 ;x  −  x 1+x  2     C B = sup x≥0  x 2 (1 + x) 2  [n][n − 1] [n +1] 2 q 2 n 1+x 1+q n x − 1  + [n] [n +1] 2 x 1+x  . (3.9) A small calculation shows that [n][n − 1] [n +1] 2 = 1 q 3 n  1 − 2+q n [n +1] + 1+q n [n +1] 2  . (3.10) Thus we have     L n  t 1+t  2 ;x  −  x 1+x  2     C B ≤ 1 q 2 n  1 − q 2 n − 2 [n +1] + 1 [n +1] 2  . (3.11) This means that condition (3.7)holdsalsoforυ = 2 and the proof is completed by the Theorem 3.2.  6 Journal of Inequalities and Applications Theorem 3.4. Let q = q n satisfies 0 <q n < 1 and let q n →1 as n→∞.IfL n is defined by (2.5), then for each x ≥ 0 and for any f ∈ H ω , the following inequality:   L n ( f ;x) − f (x)   ≤ 2ω   μ n (x)  (3.12) holds, where μ n (x) =  x 1+x  2  1 − 2 [n] [n +1] + [n][n − 1] [n +1] 2 q 2 n (1 + x)  1+q n x   + [n] [n +1] 2 x 1+x . (3.13) Proof. Since L n (1;x) = 1, we can write   L n ( f ;x) − f (x)   ≤ L n    f (t) − f (x)   ;x  . (3.14) On the other hand, f rom (3.1)and(3.3),   f (t) − f (x)   ≤ ω      t 1+t − x 1+x      ≤  1+   t/(1 + t) − x/(1 + x)   δ  ω(δ), (3.15) where we choose λ = δ −1 |t/(1 + t) − x/(1 + x)|. This inequalit y and (3.14)implythat   L n ( f ;x) − f (x)   ≤ ω(δ)  1+ 1 δ L n      t 1+t − x 1+x     ;x  . (3.16) According to t he Cauchy-Schwarz inequality, we have   L n ( f ;x) − f (x)   ≤ ω(δ)  1+ 1 δ L n      t 1+t − x 1+x     2 ;x  1/2  . (3.17) By choosing δ = μ n (x) = L n (|t/(1 + t) − x/(1 + x)| 2 ;x), we obtain desired result.  Remark 3.5. Using (3.13) and taking into consideration [n − 1]q n +1= [n]and[n +1]− [n] = q n < 1, then we have that sup x≥0 μ n (x) ≤ 1 − 2 [n] [n +1] + [n] [n +1] 2  [n − 1]q n +1  =  [n +1]− [n] [n +1]  2 ≤ 1 [n +1] 2 (3.18) holds for n large enough. Thus, if the assumptions of Theorem 3.4 hold, then, depending on the selection of q n , the rate of convergence of the operators (2.5)to f is 1/[n +1] 2 that is better than 1/(n +1) 2 , which is the rate of convergence of the BBH operators. Indeed, if we take q n = 1 − 1/(n + 2), since lim n→∞ q n n = e −1 ,therateofconvergenceofq-BBH operators t o f is exactly of order (1 − q n ) 2 = 1/(n +2) 2 that is better than 1/(n +1) 2 . A. Aral and O. Do ˘ gru 7 Now we will give an estimate concerning the rate of convergence as given in [13, 15, 16]. We define the space of general Lipschitz-type maximal functions on E ⊂ [0,∞)by W ∼ α,E as W ∼ α,E =  f :sup(1+x) α f α (x, y) ≤ M 1 (1 + y) a , x ≥ 0, y ∈ E  , (3.19) where f is bounded and continuous on [0, ∞), M is a positive constant, 0 <α≤ 1, and f α is the following function: f α (x, t) =   f (t) − f (x)   |x − t| α . (3.20) Also, let d(x,E) be the distance between x and E, that is, d(x,E) = inf  | x − y|; y ∈ E  . (3.21) Theorem 3.6. If L n is defined by (2.5), then for all f ∈ W ∼ α,E we have   L n ( f ;x) − f (x)   ≤ M  μ α/2 n (x)+2  d(x,E)  α  , (3.22) where μ n (x) defined in (3.13). Proof. Let E denote the closure of the set E. Then there exists an x 0 ∈ E such that |x − x 0 |=d(x,E), where x ∈ [0,∞). Thus, we can write   f − f (x)   ≤   f − f  x 0    +   f  x 0  − f (x)   . (3.23) Since L n is a p ositive and linear operator and f ∈ W ∼ α,E by using above inequality, then we have   L n ( f ;x) − f (x)   ≤ L n    f − f  x 0    ;x  +   f  x 0  − f (x)   ≤ ML n      t 1+t − x 0 1+x 0     α ;x  + M   x − x 0   α (1 + x) α  1+x 0  α . (3.24) If we use the classical inequality (a + b) α ≤ a α + b α for a ≥ 0,b ≥ 0, one can write     t 1+t − x 0 1+x 0     α ≤     t 1+t − x 1+x     α +     x 1+x − x 0 1+x 0     α (3.25) for 0 <α ≤ 1andt ∈ [0,∞). Consequently, we obtain L n      t 1+t − x 0 1+x 0     α ;x  ≤ L n      t 1+t − x 1+x     α ;x  +   x − x 0   α (1 + x) α  1+x 0  α . (3.26) 8 Journal of Inequalities and Applications Since L n (1;x) = 1, applying H ¨ older inequality with p = 2/α and q = 2/(2 − α), we have L n      t 1+t − x 0 1+x 0     α ;x  ≤ L n  t 1+t − x 1+x  2 ;x  α/2 +   x − x 0   α (1 + x) α  1+x 0  α . (3.27) Thus, in view of (3.24), we get (3.22).  As a particular case of Theorem 3.6,whenE = [0, ∞), the following is true. Corollary 3.7. If f ∈ W ∼ α,[0,∞) , then one has   L n ( f ;x) − f (x)   ≤ Mμ α/2 n (x), (3.28) where μ n (x) is defined in (3.13). In the following theorem, a Stancu-type formula for the remainder of q-BBH opera- tors is obtained which reduce to the formula of remainder of classical BBH operators (see [17, page 151]). Similar formula is obtained for q-Szasz Mirakyan operators in [18]. Here, [x 0 ,x 1 , ,x n ; f ] denotes the divided difference of the function f with respect to distinct points in the domain of f and can be expressed as the following formula:  x 0 ,x 1 , ,x n ; f  =  x 1 , ,x n ; f  −  x 0 , ,x n−1 ; f  x n − x 0 . (3.29) Theorem 3.8. If x ∈ (0,∞) \{[k]/[n − k +1]q k | k = 0,1,2, ,n},thenthefollowingiden- tity holds: L n ( f ;x) − f  x q  =− x n+1  n (x)  x q , [n] q n ; f  + x  n (x) n−1  k=0  x q , [k] [n − k +1]q k , [k +1] [n − k]q k+1 ; f  q k(k+1)/2−2 [n − k]  n +1 k  x k . (3.30) Proof. By using (2.5), we have L n ( f ;x) − f  x q  = 1  n (x) n  k=0  f  [k] [n − k +1]q k  − f  x q  q k(k−1)/2  n k  x k =− 1  n (x) n  k=0  x q − [k] [n − k +1]q k  x q , [k] [n − k +1]q k ; f  q k(k−1)/2  n k  x k . (3.31) Since [k] [n − k +1]  n k  =  n k − 1  , (3.32) A. Aral and O. Do ˘ gru 9 then we have L n ( f ;x) − f  x q  =− 1  n (x) n  k=0  x q , [k] [n − k +1]q k ; f  q k(k−1)/2−1  n k  x k+1 + 1  n (x) n  k=1  x q , [k] [n − k +1]q k ; f  q k(k−1)/2−k  n k − 1  x k . (3.33) Rearranging the above equality, we can write L n ( f ;x) − f  x q  =− x n+1  n (x)  x q , [n] q n ; f  q n(n−1)/2−1 + 1  n (x) n−1  k=0   x q , [k +1] [n − k]q k+1 ; f  −  x q , [k] [n − k +1]q k ; f   q k(k−1)/2−1  n k  x k+1 . (3.34) Using the equality [k +1] [n − k]q k+1 − [k] [n − k +1]q k = [n +1] [n − k][n − k +1]q k+1 , (3.35) we have the following formula for divided differences:  x q , [k] [n − k +1]q k , [k +1] [n − k]q k+1 ; f  [n +1] [n − k][n − k +1]q k+1 =  x q , [k +1] [n − k]q k+1 ; f  −  x q , [k] [n − k +1]q k ; f  , (3.36) and therefore, we obtain that the remainder formula for q-BBH can be written as (3.30).  We know that a function is convex on an interval if and only if all second-order di- vided differences of f are nonnegative. From this property and Theorem 3.8,wehavethe following result. Corollary 3.9. If f is convex and nonincreasing, then f  x q  ≤ L n ( f ;x)(n = 0,1, ). (3.37) 4. Some gener alization of L n In this section, similarly as in [13], we will define some generalization of the operators L n . 10 Journal of Inequalities and Applications We consider a sequence of linear positive oper ators as follows: L γ n ( f ;x) = 1  n (x) n  k=0 f  [k]+γ b n,k  q k(k−1)/2  n k  x k (γ ∈ R), (4.1) where b n,k satisfies the following condition: [k]+b n,k = c n , [n] c n −→ 1, for n −→ ∞ . (4.2) It is easy to check that if b n,k = [n − k +1]q k + β for any n,k and 0 <q<1, then c n = [n +1]+β and these operators turn out into Stancu-type generalization of Bleimann, Butzer, and Hahn operators based on q-integers (see [19]). If we choose γ = 0andq = 1, then the operators become the special case of Bal ´ azs-type generalization of the operators (1.1), which is given in [13]. Theorem 4.1. Let q = q n satisfies 0 <q n ≤ 1 and let q n →1 as n→∞. If f ∈ W ∼ α,[0,∞) , then the following inequality:   L γ n ( f ;x) − f (x)   C B ≤ 3M max   [n] c n + γ  α  γ [n]  α ,     1 − [n +1] c n + γ     α  [n] [n +1]  α ,1− 2 [n] [n +1] + [n][n − 1] [n +1] 2 q n  (4.3) holds for a large n. Proof. Using (2.5)and(4.1), we have   L γ n ( f ;x) − f (x)   ≤ 1  n (x) n  k=0     f  [k]+γ b n,k  − f  [k] γ + b n,k      q k(k−1)/2 n  n k  x k + 1  n (x) n  k=0     f  [k] γ + b n,k  − f  [k] [n − k +1]q k n      q k(k−1)/2 n  n k  x k +   L n ( f ;x) − f (x)   . (4.4) Since f ∈ W ∼ α,[0,∞) and by using Cor ollary 3.7 ,wecanwrite   L γ n ( f ;x) − f (x)   ≤ M  n (x) n  k=0     [k]+γ [k]+γ + b n,k − [k] γ +[k]+b n,k     α q k(k−1)/2 n  n k  x k + M  n (x) n  k=0     [k] [k]+γ + b n,k − [k] [n +1]     α q k(k−1)/2 n  n k  x k + μ α/2 n (x) [...]... functions on the semi-axis,” Koninklijke Nederlandse Akademie van Wetenschappen Indagationes Mathematicae, vol 42, no 3, pp 255–262, 1980 [9] F Altomare and M Campiti, Korovkin-Type Approximation Theory and Its Applications, vol 17 of De Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994 ¨ ¸ [10] A D Gadjiev and O Cakar, On uniform approximation by Bleimann, Butzer and Hahn operators. .. approximation of Meyer-K¨ nig and Zeller operators based g o on q-integers,” Publicationes Mathematicae Debrecen, vol 68, no 1–2, pp 199–214, 2006 [7] T Trif, “Meyer-K¨ nig and Zeller operators based on the q-integers,” Revue d’Analyse Num´rique o e et de Th´orie de l’Approximation, vol 29, no 2, pp 221–229, 2000 e [8] G Bleimann, P L Butzer, and L Hahn, “A Bernˇte˘n-type operator approximating continuous... Hahn operators on all positive semiaxis,” Transactions of Academy of Sciences of Azerbaijan Series of PhysicalTechnical and Mathematical Sciences, vol 19, no 5, pp 21–26, 1999 [11] O Agratini, “Approximation properties of a generalization of Bleimann, Butzer and Hahn operators, ” Mathematica Pannonica, vol 9, no 2, pp 165–171, 1998 [12] O Agratini, “A class of Bleimann, Butzer and Hahn type operators, ”... of operators on infinite interval,” Demonstratio Mathematica, vol 32, no 4, pp 789–794, 1999 [16] B Lenze, “Bernstein-Baskakov-Kantoroviˇ operators and Lipschitz-type maximal functions,” c in Approximation Theory (Kecskem´t, 1990), vol 58 of Colloquia Mathematica Societatis J´ nos e a Bolyai, pp 469–496, North-Holland, Amsterdam, The Netherlands, 1991 [17] U Abel and M Ivan, “Some identities for the. .. ru, On Bleimann, Butzer and Hahn type generalization of Bal´ zs operators, ” Studia g a Universitatis Babes-Bolyai Mathematica, vol 47, no 4, pp 37–45, 2002 ¸ 12 Journal of Inequalities and Applications [14] G M Phillips, Interpolation and Approximation by Polynomials, CMS Books in Mathematics/Ouvrages de Math´ matiques de la SMC, 14, Springer, New York, NY, USA, 2003 e [15] O Agratini, “Note on a... thankful to the referees for making valuable suggestions References [1] G M Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol 4, no 1–4, pp 511–518, 1997 [2] T N T Goodman, H Oruc, and G M Phillips, “Convexity and generalized Bernstein polyno¸ mials,” Proceedings of the Edinburgh Mathematical Society, vol 42, no 1, pp 179–190, 1999 [3] H Oruc and G M Phillips,... generalization of the Bernstein polynomials,” Proceedings of the ¸ Edinburgh Mathematical Society, vol 42, no 2, pp 403–413, 1999 [4] D Barbosu, “Some generalized bivariate Bernstein operators, ” Mathematical Notes, vol 1, no 1, pp 3–10, 2000 [5] A II’nskii and S Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of Approximation Theory, vol 116, no 1, pp 100–112, 2002 [6] O Do˘ ru and O... M Ivan, “Some identities for the operator of Bleimann, Butzer and Hahn involving divided differences,” Calcolo, vol 36, no 3, pp 143–160, 1999 [18] A Aral and V Gupta, The q-derivative and applications to q-Sz´ sz Mirakyan operators, ” Cala colo, vol 43, no 3, pp 151–170, 2006 [19] D D Stancu, “Approximation of functions by a new class of linear polynomial operators, ” Revue Roumaine de Math´matiques...A Aral and O Do˘ ru g [n] cn + γ ≤ α γ [n] α + 1− n × 1 [k] (x) k=0 [n + 1] n α [n + 1] cn + γ k(k qn −1)/2 11 α n k α/2 x + μn (x) k (4.5) Using the H¨ lder inequality for p = 1/α, q = 1/(1 − α), and (2.9), we obtain o γ Ln ( f ;x) − f (x) ≤ M [n] cn + γ α γ [n] α +M 1− [n + 1] cn + γ α x [n] x + 1 [n + 1] α + μα/2 (x) n (4.6) Thus, inequality (4.3) holds for x ∈ [0, ∞) Acknowledgment The authors... operators, ” Revue Roumaine de Math´matiques Pures et Appliqu´es, vol 13, pp 1173–1194, 1968 e e Ali Aral: Department of Mathematics, Kirikkale University, Yahsihan, 71450 Kirikkale, Turkey Email address: aral@science.ankara.edu.tr Og¨ n Do˘ ru: Department of Mathematics, Faculty of Sciences and Arts, Gazi University, u g Teknik Okullar, 06500 Ankara, Turkey Email address: ogun.dogru@gazi.edu.tr . generalization of Bleimann, Butzer, and Hahn operators based on q-integers (see [19]). If we choose γ = 0andq = 1, then the operators become the special case of Bal ´ azs-type generalization of the operators (1.1),. Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 79410, 12 pages doi:10.1155/2007/79410 Research Article Bleimann, Butzer, and Hahn Operators Based on the q-Integers Ali. the operators In this section, we will give the theorems on uniform convergence and rate of convergence of the operators (2.5). As in [10], for this purpose we give a space of function ω of the type