Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators

Tam metin

(1)Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91 DOI 10.1186/s13660-015-0610-y. RESEARCH. Open Access. Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators Tuba Vedi* and Mehmet Ali Özarslan *. Correspondence: tuba.vedi@emu.edu.tr Eastern Mediterranean University, Gazimagusa, Mersin 10, TRNC, Turkey. Abstract In this paper, we introduce Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators on the unbounded domain. We should note that this generalization includes various kinds of operators which have not been introduced earlier. We calculate the error of approximation of these operators by using the modulus of continuity and Lipschitz-type functionals. Finally, we give generalization of the operators and investigate their approximations. MSC: Primary 41A10; 41A25; secondary 41A36 Keywords: Schurer-Stancu and Schurer-Chlodowsky operators; modulus of continuity. 1 Introduction Generalizations of Bernstein polynomials and their q-analogues have been an intensive research area of approximation theory (see [–]). In this paper, we introduce the Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators and investigate their approximation properties. Firstly, let us recall the following notions of q-integers []. Let q > . For any integer k ≥ , the q-integer [k]q = [k] is defined by [k] =. ( – qk )/( – q), q = , k, q = ,. [] = ,. the q-factorial [k]q ! = [k]! is defined by [k]! =. [k][k – ] · · · [], k = , , , . . . , , k=. and for integers n ≥ k ≥ , q-binomial coefficients are defined by [n]! n = . [n – k]![k]! k In , the q-based Bernstein-Schurer operators were defined by Muraru [] as Bpn (f ; q; x) =. n+p–k– n+p . [k] n+p k  – qs x , f x [n] k s=.  ≤ x ≤ .. (.). k=. © 2015 Vedi and Özarslan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited..

(2) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 2 of 15. Then she obtained the Korovkin-type theorem and the order of convergence by using the modulus of continuity. She also mentioned that if q → – in (.), the operators reduce to the Schurer operators considered by Schurer [] and if p =  in (.), they contain the q-Bernstein operators []. After that, different approximation properties of the q-Bernstein-Schurer operators were studied in []. Recently, the q-Bernstein-Schurer-Kantorovich operators were defined [] as. Knp (f ; q; x). n+p–k– n+p . n+p k  – qs x x = k s= k=   + (q – )[k] [k] × + t dq t. f [n + ] [n + ] . (.). Then, the approximation rates of the q-Bernstein-Schurer-Kantorovich operators were given by means of Lipschitz class functionals and the first and the second modulus of continuity. Notice that if we choose p =  in (.), we get the q-Bernstein-Kantorovich operators which were defined by Mahmudov and Sabancıgil in []. We should also mention that in [] the authors defined a different version of q-Bernstein-Kantorovich operators, where they used the usual integral instead of q-integral in the definition. α,β In , the q-analogue of Bernstein-Schurer-Stancu operators Sn,p : C[,  + p] → C[, ] was introduced by Agrawal et al. in [] by (α,β) Sn,p (f ; q; x) =. n–k– n+p . [k] + α n k  – qs x , x f [n] + β k s= k=. where α and β are real numbers which satisfy  ≤ α ≤ β and also p is a non-negative integer. Then, Ren and Zeng introduced the Kantorovich-type-q-Bernstein-Stancu operators []. They investigated the statistical approximation properties. On the other hand, Karslı and Gupta [] introduced q-Chlodowsky operators as follows: k n–k– n x [k] n s x Cn,q (f ; x) = –q , bn f [n] bn bn k s=.  ≤ x ≤ bn ,. k=. where n ∈ N and (bn ) is a positive increasing sequence with limn→∞ bn = ∞. Then, they investigated the approximation properties of Cn,q (f ; x). Recently, the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators was introduced by the authors in [] as. (α,β) Cn,p (f ; q; x). n+p–k– n+p [k] + α x k x n+p bn = f  – qs , [n] + β bn bn k s=. (.). k=. where n ∈ N and p ∈ N and α, β ∈ R with  ≤ α ≤ β,  ≤ x ≤ bn ,  < q < , and Korovkintype approximation theorems were proved in different function spaces. Moreover, the error of approximation was computed by using the modulus of continuity and Lipschitz-type.

(3) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 3 of 15. functionals. Also, the generalization of the Chlodowsky variant of q-Bernstein-Schurer(,) (f ; q; x) gives the q-Bernstein-SchurerStancu operators was studied. Notice that Cn,p Chlodowsky operators which have not been defined yet, and additionally taking p = , we get the q-Bernstein-Chlodowsky operators []. On the other hand, from [] the first (α,β) three moments of Cn,p (f ; q; x) are as follows: (α,β) (i) Cn,p (; q; x) = ,. [n + p]x + αbn , [n] + β.

(4).  (α,β)  (iii) Cn,p t ; q; x = [n + p – ][n + p]qx + (α + )[n + p]bn x + α  bn . ([n] + β) (α,β) (ii) Cn,p (t; q; x) =. The organization of this paper is as follows. In Section , we introduce the Chlodowskytype q-Bernstein-Stancu-Kantorovich operators and calculate the moments for them. In Section , Korovkin-type theorems are proved. In Section , we obtain the rate of convergence of the approximation process in terms of the first and the second modulus of continuity and also by means of Lipschitz class functions. In Section , we study the generalization of the Kantorovich-Stancu type generalization of q-Bernstein-Chlodowsky operators and study their approximation properties.. 2 Construction of the operators The Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators are introduced as. (α,β) (f ; q; x) Kn,p. n+p–k– n+p x x k n+p  – qs := bn bn k s= k=  ( + (q – )[k])t + [k] + α bn dt, f × [n + ] + β . (.). where n ∈ N and p ∈ N and α, β ∈ R with  ≤ α ≤ β,  ≤ x ≤ bn ,  < q < . Obviously, (α,β) Kn,p is a linear and positive operator. We should notice that if we choose p = α = β =  in (α,β) (.) and taking into account that ( + (q – )[k])t = qk t, the operator Kn,p (f ; q; x) reduces to the Chlodowsky variant of the q-Bernstein Kantorovich operator []. First of all let us give the following lemma which will be used throughout the paper. (α,β). Lemma . Let Kn,p (f ; q; x) be given in (.). Then we have (α,β) (i) Kn,p (; q; x) = ,. [n + p][]x + (α + )bn , ([n + ] + β) .  [] (α,β)  (iii) Kn,p t ; q; x = [n + p – ][n + p]qx  ([n + ] + β)     q + q +   + []α [n + p]bn x + α + α + b , +   n . [n + p][] (α + )bn (α,β) (t – x); q; x = (iv) Kn,p – x+ , ([n + ] + β) ([n + ] + β) (α,β) (ii) Kn,p (t; q; x) =.

(5) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. (v). (α,β) Kn,p. (t – x) ; q; x =. Page 4 of 15. . [][n + p – ][n + p]q [][n + p] – +  x ([n + ] + β) [n + ] + β  (q + q +  + []α)[n + p] (α + ) bn x + – ([n + ] + β) [n + ] + β +. (α  + α + )bn . ([n + ] + β). Proof (i) Using (.) and Cn,q (; x) = , we get (α,β) Kn,p (; q; x) =. n+p–k– n+p x x k n+p (α,β)  – qs = Cn,p (; q; x) = . bn b k n s= k=. (.). (ii) After some calculations, we obtain (α,β) (t; q; x) Kn,p n+p–k–  n+p ( + (q – )[k])t + [k] + α x k n+p s x –q bn dt = bn bn [n + ] + β k  s= k=. n+p–k– n+p  + (q – )[k] x k [k] + α x n+p =  – qs bn + bn bn bn [n + ] + β ([n + ] + β) k s= k=. =. [n + p][]x + (α + )bn . ([n + ] + β). Whence the result. (iii) By (.) we can write. (α,β)  t ; q; x Kn,p n+p–k–   n+p x k ( + (q – )[k])t + [k] + α x n+p bn dt  – qs = bn bn [n + ] + β k  s= k= n+p–k– n+p bn (q – ) x k x n+p  – qs + q [k] =  bn b ([n + ] + β)  k n s= k= .   . +  + []α [k] + α + α + . After some calculations as in (i) and (ii), we get the desired result. (iv) Using (i) and (ii), we get. (α,β) (α,β) (α,β) (t – x); q; x = Kn,p (t; q; x) – xKn,p (; q; x) Kn,p (α + )bn [n + p][] – x+ . = ([n + ] + β) ([n + ] + β) (v) It is known that. (α,β) (α,β)  (α,β) (α,β) (t – x) ; q; x = Kn,p t ; q; x – xKn,p (t; q; x) + x Kn,p (; q; x). Kn,p Then we obtain the result directly.. .

(6) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 5 of 15. Lemma . For the second central moment, if we take supremum on [, bn ], we get the following estimate: [][n + p – ][n + p]q [][n + p]. (α,β) Kn,p (t – x) ; q; x ≤ bn +  –  ([n + ] + β) [n + ] + β  (q + q +  + []α)[n + p] (α + ) – + ([n + ] + β) [n + ] + β (α  + α + ) . + ([n + ] + β). 3 Korovkin-type approximation theorem In this section, we study Korovkin-type approximation theorems of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators. Let Cρ denote the space of all continuous functions f such that the following condition f (x) ≤ Mf ρ(x),. –∞ < x < ∞. is satisfied. It is clear that Cρ is a linear normed space with the norm f ρ = sup. –∞<x<∞. |f (x)| . ρ(x). The following theorems play an important role in our investigations. Theorem . (See []) There exists a sequence of positive linear operators Un , acting from Cρ to Cρ , satisfying the conditions lim Un (; x) – ρ = ,. (.). lim Un (φ; x) – φ ρ = ,. (.). . lim Un φ  ; x – φ  ρ = ,. (.). n→∞. n→∞. n→∞. where φ(x) is a continuous and increasing function on (–∞, ∞) such that limx→±∞ φ(x) = ±∞ and ρ(x) =  + φ  , and there exists a function f ∗ ∈ Cρ for which limn→∞ Un f ∗ – f ∗ ρ > . Theorem . (See []) Conditions (.), (.), (.) imply lim Un f – f ρ = . n→∞. for any function f belonging to the subset Cρ := {f ∈ Cρ : lim|x|→∞ Let us choose ρ(x) =  + x and consider the operators: Un(α,β) (f ; q; x) =. (α,β). Kn,p (f ; q; x) f (x). if x ∈ [, bn ], if x ∈ [, ∞)/[, bn ].. f (x) ρ(x). is finite}..

(7) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. (α,β). It should be mentioned that the operators Un we have. Page 6 of 15. act from C+x to C+x . For all f ∈ C+x ,. (α,β) (α,β) |Kn,p (f ; q; x)| |f (x)| U (f ; q; ·)+x ≤ sup + sup n    + x  bn <x<∞ + x x∈[,bn ] (α,β) |Kn,p ( + t  ; q; x)| ≤ f +x sup +  .  + x x∈[,∞). Therefore, it is clear from Lemma . that (α,β) U (f ; q; ·)+x ≤ Mf +x n provided that q := (qn ) with  < qn < , limn→∞ qn = , limn→∞ qnn = N < ∞ and bn = . We have the following approximation theorem. limn→∞ [n]  Theorem . For all f ∈ C+x  , we have. lim Un(α,β) (f ; qn ; ·) – f (·)+x = . n→∞. provided that q := (qn ) with  < qn < , limn→∞ qn = , limn→∞ qnn = N < ∞ and bn = . limn→∞ [n] Proof With help of Theorem . and Lemma .(i), (ii) and (iii), we have the following estimates, respectively: (α,β). sup. |Un. x∈[,∞). (α,β). (; qn ; x) – | |Kn = sup  + x ≤x≤bn. (; qn ; x) – | = ,  + x. (α,β). sup. |Un. x∈[,∞). (t; qn ; x) – t|  + x. [n+p][] | ([n+]+β) – |x + (t; qn ; x) – x| ≤ sup  +x ( + x ) ≤x≤bn ≤x≤bn [n + p][] (α + )bn –  + → ≤ ([n + ] + β) ([n + ] + β) (α,β). |Kn. = sup. (α+)bn ([n+]+β). and (α,β). sup. |Un. x∈[,∞). (t  ; qn ; x) – t  |  + x (α,β). (t  ; qn ; x) – x |  + x ≤x≤bn  [] [n + p – ][n + p]q  x ≤ sup –     ([n + ] + β) ≤x≤bn  + x  [n + p]bn bn q + q +    + []α + x + α + α +  ([n + ] + β)  ([n + ] + β) = sup. |Kn.

(8) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 7 of 15. [] [n + p – ][n + p]q ≤ –   ([n + ] + β)  [n + p]bn bn  q + q +   + []α + α + α + → +  ([n + ] + β)  ([n + ] + β) whenever n → ∞, since limn→∞ qn =  and. bn [n]. =  as n → ∞.. . Lemma . Let A be a positive real number independent of n and f be a continuous function which vanishes on [A, ∞]. Assume that q := (qn ) with  < qn < , limn→∞ qn = , b limn→∞ qnn = N < ∞ and limn→∞ [n]n = . Then we have (α,β) (f ; qn ; x) – f (x) = . lim sup Kn,p. n→∞ ≤x≤b. n. Proof From the hypothesis on f , one can write |f (x)| ≤ M (M > ). For arbitrary small ε > , we have ( + (q – )[k])t + [k] + α f bn – f (x) [n + ] + β  M ( + (q – )[k])t + [k] + α <ε+  bn – x δ [n + ] + β for x ∈ [, bn ] and δ = δ(ε). Using the following equality n+p–k–   n+p x k ( + (q – )[k])t + [k] + α x n+p bn – x dt  – qs bn bn [n + ] + β k  s= k=. (α,β) (t – x) ; qn ; x , = Kn,p we get by Lemma . that (α,β) (f ; qn ; x) – f (x) sup Kn,p. ≤x≤bn. M [][n + p – ][n + p]qn [][n + p] + bn – ≤ε+  δ ([n + ] + β) [n + ] + β  (q + qn +  + []α)[n + p] (α + )  (α  + α + )bn . + n – + b ([n + ] + β) [n + ] + β n ([n + ] + β). Since  < qn < , limn→∞ qn = , limn→∞ qnn = N < ∞ and limn→∞ sired result.. bn [n]. = , we have the de. Theorem . Let f be a continuous function on the semi-axis [, ∞) and lim f (x) = kf < ∞.. x→∞. Assume that q := (qn ) with  < qn < , limn→∞ qn = , limn→∞ qnn = K < ∞ and b limn→∞ [n]n = . Then (α,β) lim sup Kn,p (f ; qn ; x) – f (x) = .. x→∞ ≤x≤b. n.

(9) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 8 of 15. Proof If we apply the same techniques as in the proof of Theorem . in [] and use Lemma ., we obtain the desired result. . 4 Order of convergence In this section, we study the rate of convergence of the operators in terms of the elements of Lipschitz classes and the first and the second modulus of continuity of the function. (α,β) Firstly, we give the rate of convergence of the operators Kn,p in terms of the Lipschitz class LipM (γ ). Let CB [, ∞) denote the space of bounded continuous functions on [, ∞) endowed with the usual supremum norm. A function f ∈ CB [, ∞) belongs to LipM (γ ) ( < γ ≤ ) if the condition f (t) – f (x) ≤ M|t – x|γ. t, x ∈ [, ∞). is satisfied. Theorem . Let f ∈ LipM (γ ). Then we have (α,β) . γ / K , n,p (f ; q; x) – f (x) ≤ M δn,q (x) where [][n + p – ][n + p]q [][n + p] +  x – ([n + ] + β) [n + ] + β  (q + q +  + []α)[n + p] (α + ) (α  + α + )bn b + – x + . n ([n + ] + β) [n + ] + β ([n + ] + β). δn,q (x) =. Proof Using the monotonicity and the linearity of the operators and taking into account that f ∈ LipM (γ ), we get (α,β) K n,p (f ; q; x) – f (x) n+p n + p x k n+p–k– x  – qs = bn bn k s= k=.  ( + (q – )[k])t + [k] + α f × bn – f (x) dt [n + ] + β  n+p–k– n+p x x k n+p  – qs ≤ bn bn k s= k=  ( + (q – )[k])t + [k] + α bn – f (x) dt × f [n + ] + β  n+p–k– n+p x k n+p s x –q ≤M bn bn k s= k= γ  ( + (q – )[k])t + [k] + α bn – x dt. × [n + ] + β .

(10) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Applying Hölder’s inequality with p = (.):.  γ. and q =. Page 9 of 15.  –γ. , we have the following inequalities by. γ  ( + (q – )[k])t + [k] + α bn – x dt [n + ] + β   γ  –γ   ( + (q – )[k])t + [k] + α bn – x dt dt ≤ [n + ] + β     γ ( + (q – )[k])t + [k] + α bn – x dt . = [n + ] + β  Then we get n+p (α,β) K n,p (f ; q; x) – f (x) ≤ M.  . k=. . ( + (q – )[k])t + [k] + α bn – x [n + ] + β. . γ dt. pn,k (q; x),. n+p n+p–k– ( – qs bxn ). Again using Hölder’s inequality with where pn,k (q; x) = k= n +k p ( bxn )k s=   p = γ and q = –γ , we have (α,β) K n,p (f ; q; x) – f (x) γ n+p –γ n+p    ( + (q – )[k])t + [k] + α pn,k (q; x) bn – x dt pn,k (q; x) ≤M [n + ] + β  k=. =M. n+p . k=.  pn,k (q; x) . k=. ( + (q – )[k])t + [k] + α bn – x [n + ] + β. γ.  dt. γ / = M δn,q (x) , (α,β). where δn,q (x) := Kn,p ((t – x) ; q; x).. . Now, we give the rate of convergence of the operators by means of the modulus of continuity ω(f ; δ). Let f ∈ CB [, ∞) such that f is uniformly continuous and x ≥ . The modulus of continuity of f is given as ω(f ; δ) = max f (t) – f (x). |t–x|≤δ t,x∈[,∞). (.). It is known that for any δ >  the following inequality f (x) – f (y) ≤ ω(f ; δ) |x – y| +  δ is satisfied []. Theorem . If f ∈ CB [, ∞), we have (α,β) . K ≤ ω f ; δn,q (x) , (f ; q; x) – f (x) n,p where δn,q (x) is the same as in Theorem ... (.).

(11) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 10 of 15. Proof From monotonicity, we have n+p–k– n+p (α,β) x k n + p s x K –q n,p (f ; q; x) – f (x) ≤ bn bn k s= k=  ( + (q – )[k])t + [k] + α f dt. × – f (x) b n [n + ] + β  Now by (.) we get (α,β) K n,p (f ; q; x) – f (x) ≤. n+p  (+(q–)[k])t+[k]+α | bn – x| [n+]+β k=. δ. . +. . n+p–k– x k n+p s x –q dt × ω(f ; δ) bn bn k s= n+p–k– n+p x k n+p s x –q = ω(f ; δ) bn bn k s= k=. ω(f ; δ) + δ n+p. k=. .  ( + (q – )[k])t + [k] + α bn – x [n + ] + β . n+p–k– x k n+p s x –q dt. × bn bn k s= Then, using the Cauchy-Schwarz inequality, we have (α,β) K n,p (f ; q; x) – f (x) n+p    ( + (q – )[k])t + [k] + α ω(f ; δ) pn,k (q; x) bn – x dt ≤ ω(f ; δ) + δ [n + ] + β  k=. ×. n+p .  pn,k (q; x). k=. = ω(f ; δ) +. / ω(f ; δ)

(12) (α,β) Kn,p (t – x) ; q; x . δ. Finally, let us choose δn,q (x) the same as in Theorem .. Then we get (α,β) . ≤ ω f ; δn,q (x) . K (f ; q; x) – f (x) n,p. . Now let us denote by CB [, ∞) the space of all functions f ∈ CB [, ∞) such that f , f. ∈ CB [, ∞). Let f denote the usual supremum norm of f . The classical Peetre’s K -functional and the second modulus of smoothness of the function f ∈ CB [, ∞) are defined respectively as K(f , δ) :=. inf. . g∈CB [,∞). f – g + δ g. .

(13) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 11 of 15. and ω (f , δ) := sup f (x + h) – f (x + h) + f (x), <h<δ, x,x+h∈I. where δ > . It is known that (see [], p.) there exists a constant A >  such that √ K(f , δ) ≤ Aω (f , δ).. (.). Theorem . Let q ∈ (, ), x ∈ [, bn ] and f ∈ CB [, ∞). Then, for fixed p ∈ N , we have (α,β) . K n,p (f ; q; x) – f (x) ≤ Cω f , αn,q (x) + ω f , βn,q (x) for some positive constant C, where αn,q (x) :=. [] [] [][n + p] [n + p] –  + +  x   ([n + ] + β) [n + ] + β. (α + ) q + q +  + []α (α + )[][n + p] – bn x + + ([n + ] + β) ([n + ] + β) [n + ] + β bn α  + α +  +  ([n + ] + β) . (.). and [][n + p] (α + )bn βn,q (x) := – x + . ([n + ] + β) ([n + ] + β). (.). ∗ (f ; q; x) : CB [, ∞) → CB [, ∞) by Proof Define an auxiliary operator Kn,p. ∗ (α,β) Kn,p (f ; q; x) := Kn,p (f ; q; x) – f. . [][n + p]x + (α + )bn ([n + ] + β). + f (x).. (.). Then by Lemma . we get ∗ (; q; x) = , Kn,p. ∗ (t – x); q; x = . Kn,p. (.). For given g ∈ CB [, ∞), it follows by the Taylor formula that g(y) – g(x) = (y – x)g (x) +. y. (y – u)g. (u) du.. x. Taking into account (.) and using (.), we get ∗. K (g; q; x) – g(x) = K ∗ g(y) – g(x); q; x n,p n,p y . . ∗ ∗. = g (x)Kn,p (u – x); q; x + Kn,p (y – u)g (u) du; q; x x. y ∗ (y – u)g. (u) du; q; x . = Kn,p x.

(14) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 12 of 15. Then by (.) ∗ K (g; q; x) – g(x) n,p y (α,β) = Kn,p (y – u)g. (u) du; q; x x. [][n + p]x + (α + )bn. – u g (u) du – ([n + ] + β) x y (α,β) (y – u)g. (u) du; q; x ≤ Kn,p. [][n+p]x+(α+)bn ([n+]+β). . x. + . [][n+p]x+(α+)bn ([n+]+β). x. . [][n + p]x + (α + )bn. – u g (u) du. ([n + ] + β). Since y (α,β) . (α,β) . K ≤ g K (y – x) ; q; x (y – u)g (u) du; q; x n,p n,p x. and . [][n + p]x + (α + )bn – u g. (u) du ([n + ] + β) x [][n + p] (α + )bn  – x+ ≤ g. , ([n + ] + β) ([n + ] + β) [][n+p]x+(α+)bn ([n+]+β). . we get ∗ K (g; q; x) – g(x) n,p. (α,β) . [][n + p] (α + )bn  (y – x) ; q; x + g. – x+ . ≤ g. Kn,p ([n + ] + β) ([n + ] + β). Hence Lemma . implies that ∗ K (g; q; x) – g(x) n,p . [][n + p – ][n + p]q [][n + p] – +  x ≤ g ([n + ] + β) ([n + ] + β)  (q + q +  + []α)[n + p] (α + ) (α  + α + )bn + – bn x +  ([n + ] + β) ([n + ] + β) ([n + ] + β) (α + )bn  [][n + p] – x+ + . ([n + ] + β) ([n + ] + β). (.). ∗ (f ; q; ·) ≤ f , considering (.) and (.), for all f ∈ CB [, ∞) and g ∈ Since Kn,p  CB [, ∞), we may write from (.) that. (α,β) K n,p (f ; q; x) – f (x) ∗ (f – g; q; x) – (f – g)(x) ≤ Kn,p.

(15) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 13 of 15. ∗ [][n + p]x + (α + )bn + Kn,p (g; q; x) – g(x) + f – f (x) ([n + ] + β) [][n + p]x + (α + )bn ≤ f – g + αn,q (x)g. + f – f (x) ([n + ] + β) . . ≤  f – g + αn,q (x)g + ω f , βn,q (x) , which yields that (α,β) . K n,p (f ; q; x) – f (x) ≤ K f , αn,q (x) + ω f , βn,q (x) . ≤ Cω f , αn,q (x) + ω f , βn,q (x) , where αn,q (x) :=. [][n + p] [n + p] [] [] + +  x –   ([n + ] + β) [n + ] + β. (α + ) q + q +  + []α (α + )[][n + p] – bn x + ([n + ] + β) ([n + ] + β) [n + ] + β bn α  + α +  +  ([n + ] + β) . +. and [][n + p] (α + )bn – x + . βn,q (x) := ([n + ] + β) ([n + ] + β) . Hence we get the result.. 5 Generalization of the Kantorovich-Stancu type generalization of q-Bernstein-Chlodowsky operators In this section, we introduce a generalization of Chlodowsky-type q-Bernstein-StancuKantorovich operators. For x ≥ , consider any continuous function ω(x) ≥  and define Gf (t) = f (t).  + t . ω(t) (α,β). Let us consider the generalization of Kn,p (f ; q; x) as follows: α,β (f ; q; x) Ln,p. n+p–k– n+p x k ω(x) n + p s x  – q =  + x bn bn k s= k=   + (q – )[k] [k] + α × Gf bn + tbn dt, [n + ] + β [n + ] + β . where  ≤ x ≤ bn and (bn ) has the same properties of Chlodowsky variant of q-BernsteinSchurer-Stancu-Kantorovich operators. Notice that this kind of generalization was considered earlier for the BernsteinChlodowsky polynomials [], q-Bernstein-Chlodowsky polynomials [] and Chlodowsky variant of q-Bernstein-Schurer-Stancu operators []. Now we have the following approximation theorem..

(16) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. Page 14 of 15. Theorem . For the continuous functions satisfying lim. x→∞. f (x) = Kf < ∞, ω(x). we have α,β. lim sup. n→∞ ≤x≤b. n. |Ln,p (f ; q; x) – f (x)| = ω(x). provided that q := (qn ) with  < qn < , limn→∞ qn =  and limn→∞. bn [n]. =  as n → ∞.. Proof Obviously,. α,β (f ; q; x) – f (x) Ln,p. n+p n+p–k– x k ω(x) n + p s x  – q =  + x bn bn k s= k=. ×. . . Gf .  + (q – )[k] [k] + α bn + tbn dt – Gf (x) , [n + ] + β [n + ] + β. hence (α,β). sup ≤x≤bn. α,β |Kn,p (Gf ; q; x) – Gf (x)| |Ln,p (f ; q; x) – f (x)| = sup . ω(x)  + x ≤x≤bn. From |f (x)| ≤ Mf ω(x) and the continuity of the function f , we have |Gf (x)| ≤ Mf ( + x ) for x ≥  and Gf (x) is a continuous function on [, ∞). Using Theorem ., we get the desired result. α,β. (α,β). Finally note that taking ω(x) =  + x , the operator Ln,p (f ; q; x) reduces to Kn,p (f ; q; x). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Received: 30 October 2014 Accepted: 24 February 2015 References 1. Erençin, A, Ba¸scanbaz-Tunca, G, Ta¸sdelen, F: Kantorovich type q-Bernstein-Stancu operators. Stud. Univ. Babe¸s-Bolyai, Math. 57(1), 89-105 (2012) 2. Agrawal, PN, Gupta, V, Kumar, SA: On a q-analogue of Bernstein-Schurer-Stancu operators. Appl. Math. Comput. 219, 7754-7764 (2013) 3. Barbosu, D: Schurer-Stancu type operators. Stud. Univ. Babe¸s-Bolyai, Math. XLVIII(3), 31-35 (2003) 4. Barbosu, D: A survey on the approximation properties of Schurer-Stancu operators. Carpath. J. Math. 20, 1-5 (2004) 5. Büyükyazıcı, ˙I: On the approximation properties of two dimensional q-Bernstein-Chlodowsky polynomials. Math. Commun. 14(2), 255-269 (2009) 6. Büyükyazıcı, ˙I, Sharma, H: Approximation properties of two-dimensional q-Bernstein-Chlodowsky-Durrmeyer operators. Numer. Funct. Anal. Optim. 33(2), 1351-1371 (2012) 7. Chlodowsky, I: Sur le développement des fonctions définies dans un intervalle infini en séries de polynomes de M.S. Bernstein. Compos. Math. 4, 380-393 (1937) 8. DeVore, RA, Lorentz, GG: Constructive Approximation. Springer, Berlin (1993) 9. Gadjiev, AD: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P.P. Korovkin. Dokl. Akad. Nauk SSSR 218(5), 1001-1004 (1974). English translation in Sov. Math. Dokl. 15(5), 1433-1436 (1974) 10. Gadjiev, AD: Theorems of the type of P.P. Korovkin’s theorems. Mat. Zametki 20(5), 781-786 (1976).

(17) Vedi and Özarslan Journal of Inequalities and Applications (2015) 2015:91. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.. Page 15 of 15. ˙Ibikli, E: On Stancu type generalization of Bernstein-Chlodowsky polynomials. Mathematica 42(65), 37-43 (2000) ˙Ibikli, E: Approximation by Bernstein-Chlodowsky polynomials. Hacet. J. Math. Stat. 32, 1-5 (2003) Karslı, H, Gupta, V: Some approximation properties of q-Chlodowsky operators. Appl. Math. Comput. 195, 220-229 (2008) Lupa¸s, AA: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, vol. 9, pp. 85-92. University of Cluj-Napoca, Cluj-Napoca (1987) Özarslan, MA: q-Szasz Schurer operators. Miskolc Math. Notes 12, 225-235 (2011) Phillips, GM: On generalized Bernstein polynomials. In: Numerical Analysis, vol. 98, pp. 263-269. World Scientific, River Edge (1996) Phillips, GM: Interpolation and Approximation by Polynomials. Springer, New York (2003) Stancu, DD: Asupra unei generalizari a polinoamelor lui Bernstein (On generalization of the Bernstein polynomials). Stud. Univ. Babe¸s-Bolyai, Ser. Math.-Phys. 14(2), 31-45 (1969) Yali, W, Yinying, Z: Iterates properties for q-Bernstein-Stancu operators. Int. J. Model. Optim. 3(4), 362-368 (2013). doi:10.7763/IJMO.2013.V3.299 Kac, V, Cheung, P: Quantum Calculus. Springer, Berlin (2002) Muraru, CV: Note on q-Bernstein-Schurer operators. Stud. Univ. Babe¸s-Bolyai, Math. 56, 489-495 (2011) Schurer, F: Linear Positive Operators in Approximation Theory. Math. Inst., Techn. Univ. Delf Report (1962) Vedi, T, Özarslan, MA: Some properties of q-Bernstein-Schurer operators. J. Appl. Funct. Anal. 8(1), 45-53 (2013) Özarslan, MA, Vedi, T: q-Bernstein-Schurer-Kantorovich operators. J. Inequal. Appl. 2013, 444 (2013). doi:10.1186/1029-242X-2013-444 Mahmudov, NI, Sabancıgil, P: Approximation theorems of q-Bernstein-Kantorovich operators. Filomat 27(4), 721-730 (2013). doi:10.2298/FIL1304721M Mahmudov, NI, Kara, M: Approximation theorems for generalized complex Kantorovich-type operators. J. Appl. Math. (2012). doi:10.1155/2012/454579 Zeng, X: Some statistical approximation properties of Kantorovich-type-q-Bernstein-Stancu operators. J. Inequal. Appl. (2014). doi:10.1186/1029-242X-2014-10 Vedi, T, Özarslan, MA: Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. J. Inequal. Appl. (2014). doi:10.1186/1029-242X-2014-189.

(18)