Szász-Baskakov type operators based on q-integers

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1 Agrawal et al. Joural of Iequalities ad Applicatios :441 R E S E A R C H Ope Access Szász-Baskakov type operators based o q-itegers Purshottam N Agrawal 1 HaruKarsli 2 ad Meeu Goyal 1* * Correspodece: meeu.goyaliitr@yahoo.com 1 Departmet of Mathematics Idia Istitute of Techology Roorkee Roorkee Idia Full list of author iformatio is available at the ed of the article Abstract I the preset paper we itroduce the q-aalog of the Stacu variat of Szász-Baskakov operators. We establish the momets of the operators by usig the q-derivatives ad the prove the basic covergece theorem. Net the Voroovskaja type theorem ad some direct results for the above operators are discussed. Also the rate of covergece ad weighted approimatio by these operators i terms of modulus of cotiuity are studied. The we obtai a poit-wise estimate usig the Lipschitztypemaimalfuctio. Lastly westudythea-statistical covergece of these operators ad also i order to obtai a better approimatio we study a Kig type modificatio of the above operators. MSC: 41A25; 26A15; 40A35 Keywords: Szász-Baskakov operators; rate of covergece; modulus of cotiuity; weighted approimatio; poit-wise estimates; Lipschitz type maimal fuctio; statistical covergece 1 Itroductio I recet years there has bee itesive research o the approimatio of fuctios by positive liear operatorsitroducedby makig use of q-calculus.lupas [1] ad Phillips [2] pioeered the study i this directio by itroducig q-aalogs of the well-kow Berstei polyomials. Derrieic [3] discussed modified Berstei polyomials with Jacobi weights. Gupta [4] ad Gupta ad Hepig [5] proposedtheq-aalogs of usual ad discretely defied Durrmeyer operators ad studied their approimatio properties. I [6] Stacu itroduced the positive liear operators P β : C[0 1] C[0 1] by modifyig the Berstei polyomials as P β f ; = k p k f β k=0 where p k = k k 1 k [0 1] is the Berstei basis fuctio ad β are ay two real umbers satisfyig 0 β. RecetlyBuyukyazici[7] itroducedthestacu type geeralizatio of certai q-baskakov operatorsad studied some local direct results for these operators. Subsequetly several authors cf. [8 11] etc. havecosideredsucha modificatio for some other sequeces of positive liear operators Agrawal et al.; licesee Spriger. This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited.

2 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 2 of 18 To approimate Lebesgue itegrable fuctios defied o [0 Gupta et al. [12] itroduced the followig positive liear operators: M f ; = q ν ν=0 1 Bν1 ν 0 b ν tf t dt 1.1 ν where b ν = q 1 ν1 ν =e [0 ad studied the asymptotic ν! approimatio ad error estimatio i simultaeous approimatio. Subsequetly for f C γ [0 :=f C[0 : f t M f 1 t γ forsomem f >0γ >0 Gupta[13] itroduced the followig operators: S f ; = q ν ν=1 0 b ν 1 tf t dt e f by cosiderig the value of the fuctio at zero eplicitly ad studied a estimate of error i terms of the higher order modulus of cotiuity i simultaeous approimatio for a liear combiatio of the operators 1.2 itroduced by May [14]. Later o Gupta ad Noor [15] discussed some direct results i a simultaeous approimatio for the operators 1.2. I [16] Aral itroduced Szász-Mirakja operators based o q-itegers ad established some direct resultsad gave two represetatios ofthe rth q-derivative of these operators. AraladGupta [17] studied thebasic covergecetheoremforthe rth order q-derivatives a Voroovskaja type theorem ad some more properties of these operators. Agratii ad Doǧru [18] costructed Szász-Kig type operators ad ivestigated the statistical covergece ad the rate of local ad global covergece for fuctios with a polyomial growth.örkcü ad Doǧru [19] proposed two differet modificatios of q-szász-mirakja Katorovich operators ad obtaied the rate of covergece i terms of the modulus of cotiuity. I [20]they itroducedkatorovichtype geeralizatio of q-szász-mirakja operators ad discussed their A-statistical approimatio properties. f Let the space C γ [0 be edowed with the orm f γ = sup t [0 1t γ the for f C γ [0 0 < q <10 β ad each positive iteger we itroduce the followig Stacu type modificatio of the operators 1.2basedoq-itegers: B β f ; = /A q q ν q q ν 1 ν [] q t b ν 1 q tf ν=1 0 [] q β e q []q f [] q β d q t 1.3 where q ν q = e q [] q [] ν q ν [ν] q! ad b ν q t= t ν q νν 1/2 1t ν1 B q ν1. For = β =0wedeoteB β f ; byb f ;. Clearly if q 1 ad = β = 0 the operators defied by 1.3 reduce to the operators give by 1.2. The purpose of the preset paper is to study the basic covergece theorem Voroovskaja type asymptotic formula local approimatio rate of covergece weighted approimatio poit-wise estimatio ad A-statistical covergece of the operators 1.3. Fur-

3 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 3 of 18 ther to obtai a better approimatio we also propose a modificatio of these operators by usig a Kig type approach. We recall that the q-aalogofbetafuctioofsecodkid[21 22]isdefiedby /A t 1 B q t s=ka t d 0 1 q ts q 1.4 where K t= 1 1 t 1 1 t q 1 1 t q ada bs q = s 1 i=0 a qi b s Z. I particular for ay positive itegers mwehave K =q 1 2 K0=1 1.5 ad B q m = Ɣ qmɣ q Ɣ q m. 2 Momet estimates Lemma 1 For B t m ; m =012oe has 1 B 1; =1; 2 B t; = [] q [ 1] q for >1; 3 B t 2 ; = q[] 2 q [ 1] q [ 2] q 2 [2] q [] q q[ 1] q [ 2] q for >2. Proof We observe that B are well defied for the fuctios 1 t t 2.Thusforevery [0 usig 1.4ad1.5 we obtai B 1; = = /A q ν q q ν 1 b ν 1 q t d q t e q []q ν=1 q ν q =1. ν=0 Net for f t=tagaiapplyig 1.4 ad1.5 weget 0 B t; = /A q ν q q ν 1 b ν 1 q tq ν td q t ν=1 0 = [] q [ 1] q. Proceedig similarly we have B t 2 ; = [2] q [] q q[] 2 q 2 q[ 1] q [ 2] q [ 1] q [ 2] q by usig [ν 2] q =[2] q q 2 [ν] q. Lemma 2 For the operators B β f ; as defied i 1.3 the followig equalities hold: 1 B β 1; =1;

4 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 4 of 18 2 B β t; = [] 2 q [] q β[ 1] q [] q for >1; β 3 B β t 2 ; = [] q [] q β 2 q[]2 q 2[ 2] q [ 1] q [ 2] q 2 [2] q [] q q[ 1] q [ 2] q [] q β 2 for >2. Proof This lemma is a immediate cosequece of Lemma 1. Hece the details of its proof are omitted. Lemma 3 For f C B [0 the space of all bouded ad uiform cotiuous fuctios o [0 edowed with orm f = sup f : [0 oe has B β f ; f. Proof I view of 1.3 ad Lemma 2 the proofofthislemmaeasilyfollows. Remark 1 For every q 0 1 we have ad B β [] 2 q [] q β[ 1] q [ 1] q t ; = >1 [] q β[ 1] q B β t 2 ; q[] 4 q = 1 1 [] q β 2 [ 1] q [ 2] q =: γ [2] q [] 3 q q[] q β 2 [ 1] q [ 2] q 2 [] q β 2[] 2 q [] q β 2 [ 1] q 2 2 [] q β 2 >2 say. 3 Mai results Theorem 1 Let 0<q <1ad A >0.The for each f C γ [0 the sequece B β f ; coverges to f uiformly o [0 A] if ad oly if lim q =1. Proof First we assume that lim q =1. We have to show that B β f ; coverges to f uiformly o [0 A]. From Lemma 2weseethat B β 1; 1 B β t; uiformly o [0 A]as. B β t 2 ; 2 Therefore the well-kow property of the Korovki theorem implies that B β f ; coverges to f uiformly o [0 A] provided f C γ [0. We show the coverse part by cotradictio. Assume that q does ot coverge to 1. The it must cotai a subsequece q k such that q k 0 1 q k a [0 1 as k.

5 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 5 of 18 Thus 1 [ k s] qk = from Lemma 2wehave 1 q k 1 q k k s 1 a ask.choosig = k q = q k i B β t 2 ; B β t 2 ; a 21 a2 1 a aβ 2 a1 1 aβ 1 a aβ 2 2 as k which leads us to a cotradictio. Hece lim q = 1. This completes the proof. Theorem 2 Voroovskaja type theorem Let f C γ [0 ad q 0 1 be a sequece such that q 1 ad q 0 as. Suppose that f eists at a poit [0 the we have lim [] q B β f ; f = βf 1 1 f. Proof By Taylor s formula we have f t=f t f 1 2 f t 2 rt t where rt is the Peao form of the remaider ad lim t rt =0. Applyig B β f to the both sides of 3.1 we have [] q B β f ; f =[] q f B β 1 t ; 2 [] q f B β t 2 ; [] q B β t 2 rt ;. I view of Remark 1wehave ad lim [] q B β t ; = β 3.2 lim [] q B β t 2 ; = Now we shall show that [] q B β rt t 2 ; 0 whe. By usig the Cauchy-Schwarz iequality we have B β rt t 2 ; B β r2 t ; B β t 4 ;. 3.4 We observe that r 2 =0adr 2 C γ [0. The it follows from Theorem 1 that lim Bβ r 2 t ; = r 2 =0 3.5

6 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 6 of 18 i view of the fact that B β t 4 ; =O 1 [] 2 q. Now from 3.4ad3.5 we get lim Bβ rt t 2 ; =0 3.6 ad from ad 3.6 we get the required result. Theorem 3 Voroovskaja type theorem Let f C γ [0 ad q 0 1 be a sequece such that q 1 ad q 0 as. If f eists o [0 the lim [] q B β f ; f = βf 1 1 f holds uiformly o [0 A] where A >0. Proof Let [0 A]. The remaider part of the proof of this theorem is similar to that of the proof of the previous theorem. So we omit it. 3.1 Local approimatio For C B [0 let us cosider the followig K-fuctioal: K 2 f δ= if g W 2 f g δ g 3.7 where δ >0adW 2 = g C B [0 :g C B [0. By[23] thereeistsaabsolute costat C >0suchthat K 2 f δ Cω 2 f δ 3.8 where ω 2 f δ= sup sup 0<h δ [0 is the secod order modulus of smoothess of f.by ωf δ= sup sup 0<h δ [0 f 2h 2f hf 3.9 f h f we deote the usual modulus of cotiuity of f C B [0. Theorem 4 Let f C B [0 ad q 0 1. The for every [0 ad 2 we have B β f t; f Cω2 f δ ω where C is a absolute costat ad δ = B β f [] qq 1 β[ 1] q [ 1] q [] q β[ 1] q t 2 ; [ []q q 1 ] β[ 1] q [ 1] 2 1/2 q. [] q β[ 1] q

7 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 7 of 18 Proof For [0 we cosider the auiliary operators B β B β f ; =Bβ f ; f [] 2 q [] q β[ 1] q From Lemma 2 we observe that the operators B β fuctios. Hece defied by f [] q β are liear ad reproduce the liear B β t ; = Let g W 2. By Taylor s theorem we have gt=gg t Applyig B β B β t t ug u du t [0. to the both sides of the above equatio ad usig 3.11 we have g; =gbβ Thus by 3.10weget t β B g; g t B β t u g u du; [ B β [] 2 q []qβ[ 1]q []qβ t ug u du;. [] 2 q [] q β[ 1] q [] q β u g u du t 2 ; [] 2 q 2 ] g [] q β[ 1] q [] q β δ 2 g O other had by 3.10 ad Lemma 3wehave B β f ; B β f ; 2 f 3 f Usig 3.12ad3.13 i3.10 we obtai B β f ; f B β f g; f g B β g; g f [] 2 q f [] q β[ 1] q [] q β 4 f g δ 2 g [] 2 f q [ 1] q f [] q β[ 1]. q Hece takig the ifimum o the right had side over all g W 2 ad usig 3.8 we get the required result.

8 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 8 of 18 Theorem 5 Let f C 2 [0 q 0 1 ad ω a1 f δ be its modulus of cotiuity o the fiite iterval [0a 1] [0 where a >0.The for every >2 B β f ; f 4Mf 1a 2 γ 2ω a1 f γ where γ is defied i Remark 1. Proof From [24] for [0 a]adt [0 we get f t f 4Mf 1a 2 t 2 1 t ω a1 f δ δ >0. δ Thus by applyig the Cauchy-Schwarz iequality we have B β f ; f 4M f 1a 2 B β t 2 ; ω a1 f δ 1 1 B β δ t 2 ; 2 1 =4M f 1a 2 γ 2ω a1 f γ o choosig δ = γ. This completes the proof of the theorem. 3.2 Weighted approimatio Let C 2 [0 :=f C 2[0 :lim f 1 2 <. Throughout the sectio we assume that q is a sequece i 0 1 such that q 1. Theorem 6 For each f C2 [0 we have lim B β f f 2 =0. Proof Makig use of the Korovki type theorem o weighted approimatio [25] we see that it is sufficiet to verify the followig three coditios: lim B β t k ; k 2 =0 k = Sice B β 1; = 1 the coditio i 3.14holdsfork =0. By Lemma 2wehavefor >1 B β t; 2 [] q q 1 β[ 1] q [] q β[ 1] q [] q q 1 β[ 1] q [] q β[ 1] q sup [0 1 2 [] q β [] q β which implies that the coditio i 3.14holds for k =1. sup [

9 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 9 of 18 Similarly we ca write for >2 B β t 2 ; 2 2 q [] 2 q 2 1 [ 1] q [ 2] q [ 1] q [2] q [] q 2 q [ 1] q [ 2] q [] q β 2 which implies that lim B β t 2 ; 2 2 =03.14holdsfork =2. Now we preset a weighted approimatio theorem for fuctios i C2 [0. Such type of results are proved i [26] for classical Szász operators. Theorem 7 For each f C2 [0 ad >0we have lim sup [0 B β f ; f =0. Proof Let 0 [0 be arbitrary but fied. The B β sup f ; f sup [ β B 0 f ; f sup β B > 0 B β f f C[00 ] f 2 sup f ; f β B > 0 1 t 2 ; f sup > Sice f f f we have sup >0 f Let ɛ > 0 be arbitrary. We ca choose 0 to be so large that f < ɛ I view of Theorem 1we obtai B β f 2 lim 1 t 2 ; 1 2 = f 1 2 = f f < ɛ Usig Theorem 5we casee that thefirst termoftheiequality 3.15 implies that B β f f C[00 ] < ɛ as Combiig we get the desired result. For f C2 [0 the weighted modulus of cotiuity is defied as 2 f δ= sup 00<h δ f h f 1 h 2.

10 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 10 of18 Lemma 4 [27] If f C2 [0 the i 2 f δ is mootoe icreasig fuctio of δ ii lim δ 0 2 f δ=0 iii for ay λ [0 2 f λδ 1 λ 2 f δ. Theorem 8 If f C2 [0 the for sufficietly large we have B β f ; f K 1 2λ 2 f δ [0 where λ 1 δ = ma β γ β γ beig q [] 4 q 2[ 2] q [] 2 q [ 1] q [ 2] q [] q β [] q β 2 2[] 3 q q [ 1] q [ 2] q [] q β 2 ad respectively ad K is a positive costat idepedet of f ad. Proof From the defiitio of 2 f δ ad Lemma 4wehave f t f 1 t t 2 1 t δ t δ := φ t 1 1 δ ψ t 2 f δ 2 f δ 2 f δ where φ t=12 t 2 ad ψ t= t.theweobtai B β f ; f B β φ ; 1 B β δ φ ψ ; 2 f δ. Now applyig the Cauchy-Schwarz iequality to the secod term o the right had side we get B β f ; f B β φ ; 1 B β δ φ 2 ; B β ψ 2 ; 2 f δ From Lemma Bβ 1t 2 ; = where C 1 is a positive costat. 1 q 1 [] 4 q 2[ 2] q [] 2 q 2 2 [ 1] q [ 2] q [] q β [2] q [] 3 q q [ 1] q [ 2] q [] q β [] q β C 1 forsufficietlylarge 3.20

11 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 11 of18 From 3.20 there eists a positive costat K 1 such that B β φ ; K for sufficietly large. 1 Proceedig similarly B β t 4 ; 1C 2 forsufficietlylarge wherec 2 is a positive costat. So there eists a positive costat K 2 suchthat [0 ad is large eough. Also we get B β ψ 2 ; q [] 4 q = 2[ 2] q [] 2 q [ 1] q [ 2] q [] q β 1 2[] 2 q 2 2[] 3 q q [ 1] q [ 2] q [] q β 2 2 β γ. B β φ 2 ; K 21 2 where [] q β[ 1] q 2 [] q β 2 2 [] q β 2 Hece from 3.19 we have B β f ; f 1 2 K 1 1 K 2 δ 2 β γ 2 f δ. If we take δ = ma β γ theweget B β f ; f 1 2 K 1 K f δ K 3 1 2λ 2 f δ for sufficietly large ad [0. Hece the proof is completed. 3.3 Poit-wise estimates I this sectio we establish some poit-wise estimates of the rate of covergece of the operators B β. First we give the relatioship betwee the local smoothess of f ad local approimatio. We kow that a fuctio f C[0 isilip M η o E η 0 1] E be ay bouded subset of the iterval [0 if it satisfies the coditio f t f M t η t [0 ad E where M is a costat depedig oly o ad f. Theorem 9 Let f C[0 Lip M η E [0 ad 0 1]. The we have B β f ; f M γ η/2 2dη E [0 where M is a costat depedig o η ad f ad d E is the distace betwee ad E defied as d E=if t : t E.

12 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 12 of18 Proof Let E be the closure of E i [0. The there eists at least oe poit 0 E such that d E= 0. By our hypothesis ad the mootoicity of B β weget B β f ; f B β f t f 0 ; B β f f 0 ; M B β t 0 η ; 0 η M B β t η ; 2 0 η. Now applyig Hölder s iequality with p = 2 η ad 1 q =1 1 p weobtai B β f ; f M [ B β t 2 ; ] η/2 2d η E from which the desired result immediate. Net we obtai the local direct estimate of the operators defied i 1.3 usig the Lipschitz-type maimal fuctio of order η itroduced by Leze [28]as ω η f = sup t t [0 f t f t η [0 adη 0 1] Theorem 10 Let f C B [0 ad 0<η 1. The for all [0 we have B β f ; f ωη f γ η/2. Proof From 3.21 we have B β f ; f ω η f B β t η ;. Applyig Hölder s iequality with p = 2 η ad 1 q =1 1 p weget B β f ; f ω η f B β t 2 ; η 2 ω η f γ η/2. Thus the proof is completed. 4 Statistical covergece Let A =a k be a o-egative ifiite summability matri. For a give sequece := the A-trasform of deoted by A :A is defied as A = a k k k=1 provided the series coverges for each.aissaidtoberegulariflim A = L wheever lim = L.The = is said to be A-statistically coverget to Li.e.st A - lim = L

13 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 13 of18 if for every ɛ >0lim k: k L ɛ a k =0.IfwereplaceA by C 1 the A isacesaromatriof order oe ad A-statistical covergece is reduced to the statistical covergece. Similarly if A = I the idetity matri the A-statistical covergece is called ordiary covergece. Kolk [29] proved that statistical covergece is better tha ordiary covergece. I this directio sigificat cotributios have bee made by cf. [ ] etc.. Let q 0 1 be a sequece such that st A - lim q =1 st A - lim q = a a <1 ad st A- lim 1 [] q = Theorem 11 Let A =a k be a o-egative regular summability matri ad q be a sequece satisfyig 4.1. The for ay compact set K [0 ad for each fuctio f CK we have st A - lim B β f ; f =0. Proof Let 0 = ma K. From Lemma 2 st A - lim B β e 0 ; e 0 =0.Agaiby Lemma 2wehave supb β e 1 ; e 1 K For ɛ > 0 let us defie the followig sets: [] 2 q 1 [] q β[ 1] 0 q [] q β. G := k : B β kq k e 1 ; e 1 ɛ [k] G 1 := k : 2 q k 1 [k] qk β[k 1] ɛ qk 2 G 2 := k : [k] qk β ɛ 2 which implies that G G 1 G 2 ad hece for all Nweobtai a k a k a k. k G 1 k G 2 k G Hece takig the limit as wehavest A - lim B β e 1 ; e 1 =0. Similarly by usig Lemma 2 we have sup B β e 2 ; e 2 q [] 4 q 2[ 2] q K [ 1] q [ 2] q [] q β 1 2 Now let us defie the followig sets: 2 0 [2] q [] 3 q q [ 1] q [ 1] q [] q β [] q β. 2 M := k : B β kq k e 2 ; e 2 ɛ

14 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 14 of18 M 1 := k : M 2 := k : M 3 := k : q k [k] 4 q k 2[k 2] qk [k 1] qk [k 2] qk [k] qk β 1 2 ɛ 3 [2] qk [k] 3 q k q k [k 1] qk [k 1] qk [k] qk β ɛ [k] qk β 2 ɛ 3 The we obtai M M 1 M 2 M 3 which implies that a k a k a k a k. k M k M 1 k M 2 k M 3 Thus as we get st A - lim B β e 2 ; e 2 = 0. This completes the proof. Theorem 12 Let A =a k be a o-egative regular summability matri ad q be a sequece i 0 1 satisfyig 4.1. Let the operators B β N be defied as i 1.3. The for each fuctio f C 2 [0 we have st A - lim B β f ; f ρ =0 where ρ=1 2λ λ >0. Proof From [31p.191Theorem3]itissufficiettoprovethatst A - lim B β e i ; e i 2 = 0 where e i = i i =012. From Lemma 2 st A - lim B β e 0 ; e 0 2 =0holds. Agai usig Lemma 2wehave B β e 1 ; e 2 [] 1 sup 2 q [ [] q β[ 1] 1 q 1 2 [] q β [] 2 q 1 [] q β[ 1] q [] q β. 4.2 For each ɛ > 0 let us defie the followig sets: G := k : B β kq k e 1 ; e 1 ɛ [k] G 1 := k : 2 q k 1 [k] qk β[k 1] ɛ qk 2 G 2 := k : [k] qk β ɛ 2 which yields G G 1 G 2 i view of 4.2 ad therefore for all Nwehave a k a k a k. k G 1 k G 2 k G

15 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 15 of18 Hece o takig the limit as st A - lim B β e 1 ; e 1 2 = 0. Proceedig similarly B β e 2 ; e 2 2 Now let us defie the followig sets: q [] 4 q 2[ 2] q [ 1] q [ 2] q [] q β 1 2 [2] q [] 3 q q [ 1] q [ 1] q [] q β 2 2 [] q β 2. M := k : B β kq k e 2 ; e 2 ɛ q M 1 := k : k [k] 4 q k 2[k 2] qk [k 1] qk [k 2] qk [k] qk β 1 2 ɛ 3 M 2 := k : M 3 := k : [2] qk [k] 3 q k q k [k 1] qk [k 1] qk [k] qk β ɛ [k] qk β 2 ɛ 3 The we obtai M M 1 M 2 M 3 which implies that a k a k a k a k. k M k M 1 k M 2 k M 3 Hece takig the limit as we get st A - lim B β e 2 ; e 2 2 =0.Thiscompletes the proof of the theorem. 5 Better estimates It is well kow that the classical Berstei polyomials preserve costat as well as liear fuctios. To make the covergece faster Kig [38] proposed a approach to modify the Berstei polyomials so that the sequece preserves test fuctios e 0 ad e 2 where e i t =t i i =012. As the operator B β f ; defiedi1.3 reproduces oly costat fuctios this motivated us to propose the modificatio of this operator so that it ca preserve costat as well as liear fuctios. The modificatio of the operators give i 1.3isdefiedas B β f ; = q ν r q /A q q ν 1 ν [] q t b ν 1 q tf ν=1 0 [] q β e q []q r q f [] q β d q t where r q = [] qβ[ 1] q [ 1] q [] 2 q for I =[ [] q β ad >1. Lemma 5 For each I by simple computatios we have 1 B β 1; =1; 2 B β t; =;

16 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 16 of18 3 B β t 2 ; = [ 1] qq[] 2 q 2[ 2] q 2 [2] q[] 3 q 2q[ 1] qq[] 2 q 2[ 2] q [ 2] q [] 2 q q[] q β[ 2] q [] 2 q 2 [] 2 q q[ 1] q[ 2] q 2 3 [ 1] q [ 2] q [] q β 2 [] 2 q[ 2] q. Cosequetly for each I we have the followig equalities: B β B β t ; =0 t 2 ; = []2 q q[ 1] q [ 2] q 2[ 1] q [ 2] q [ 2] q [] 2 q [2] q[] 3 q 2q[ 1] qq[] 2 q 2[ 2] q q[] q β[ 2] q [] 2 q 2 [] 2 q q[ 1] q [ 2] q 2[ 1] q [ 2] q [] q β 2 [] 2 q [ 2] q =: ξ say. Theorem 13 Let f C B [0 ad I. The there eists a positive costat C such that B β f ; f Cω 2 f ξ where ξ is give by 5.1. Proof Let g W 2 I ad t [0. Usig Taylor s epasio we have gt=gt g Applyig B β B β t t ug u du. o both sides ad usig Lemma 5weget g; g= B β t t ; g B β Obviously we have t t ug u du t 2 g. Therefore B β g; g B β t 2 ; g = ξ g. Sice B β f ; f weget t ug u du;. B β f ; f B β f g; f g B β g; g 2 f g ξ g. Fially takig the ifimum over all g W 2 ad usig weobtai B β f ; f Cω2 f ξ which proves the theorem.

17 Agrawal et al. Joural of Iequalities ad Applicatios :441 Page 17 of18 Competig iterests The authors declare that they have o competig iterests. Authors cotributios All authors cotributed equally to the writig of this paper. All authors read ad approved the fial mauscript. Author details 1 Departmet of Mathematics Idia Istitute of Techology Roorkee Roorkee Idia. 2 Departmet of Mathematics Faculty of Sciece ad Arts Abat Izzet Baysal Uiversity Bolu Turkey. Ackowledgemets The authors are etremely grateful to the reviewers for a careful readig of the mauscript ad makig valuable suggestios leadig to a better presetatio of the paper. The last author is thakful to the Coucil of Scietific ad Idustrial Research Idia for fiacial support to carry out the above research work. Received: 18 Jue 2014 Accepted: 1 October 2014 Published: 03 Nov 2014 Refereces 1. LupasA:A q-aalogue of the Berstei operator. I: Semiar o Numerical ad Statistical Calculus Cluj-Napoca 1987 vol. 9 pp Uiv. Babeş-Bolyai Cluj-Napoca Phillips GM: Berstei polyomials based o the q-itegers. The heritage of P. L. Chebyshev: a Festschrift i hoour of the 70th birthday of Prof. T. J. Rivli. A. Numer. Math Derrieic MM: Modified Berstei polyomials ad Jacobi polyomials i q-calculus. Red. Circ. Mat. Palermo Suppl Gupta V: A ote o q-baskakov-szász operators. Lobachevskii J. Math Gupta V Hepig W: The rate of covergece of q-durrmeyer operators for 0 < q < 1. Math. Methods Appl. Sci Stacu DD: Approimatio of fuctios by a ew class of liear polyomial operators. Rev. Roum. Math. Pures Appl Buyukyazici I: O Stacu type geeralizatio of q-baskakov operators. Math. Comput. Model Agrawal PN Gupta V Sathish Kumar A: O q-aalogue of Berstei-Schurer-Stacu operators. Appl. Math. Comput Agrawal PN Sathish Kumar A Siha TAK: Stacu type geeralizatio of modified Schurer operators based o q-itegers. Appl. Math. Comput Gupta V Karsli H: Some approimatio properties by q-szász-mirakya-baskakov-stacu operators. Lobachevskii J. Math Verma DK Gupta V Agrawal PN: Some approimatio properties of Baskakov-Durrmeyer-Stacu operators. Appl. Math. Comput Gupta V Srivastava GS Sahai A: O simultaeous approimatio by Szász-beta operators. Soochow J. Math Gupta V: Error estimatio for mied summatio-itegral type operators. J. Math. Aal. Appl May CP: Saturatio ad iverse theorems for combiatios of a class of epoetial type operators. Ca. J. Math Gupta V Noor MA: Covergece of derivatives for certai mied Szász-beta operators. J. Math. Aal. Appl Aral A: A geeralizatio of Szász-Mirakja operators based o q-itegers. Math. Comput. Model AralAGuptaV:The q-derivative ad applicatios to q-szász-mirakya operators. Calcolo Agratii O Doǧru O: Weighted approimatio by q-szász-kig type operators. Taiwa. J. Math ÖrkcüMDoǧru O: q-szász-mirakja Katorovich type operators preservig some test fuctios. Appl. Math. Lett ÖrkcüMDoǧru O: Statistical approimatio of a kid of Katorovich type q-szász-mirakja operators. Noliear Aal Aral A Gupta V Agarwal RP: Applicatios of q-calculus i Operator Theory. Spriger Berli Kac V Cheug P: Quatum Calculus. Spriger New York DeVore RA Loretz GG: Costructive Approimatio. Spriger Berli Ibikli E Gadjieva EA: The order of approimatio of some ubouded fuctio by the sequeces of positive liear operators. Turk. J. Math Gadjieva AD: A problem o the covergece of a sequece of positive liear operators o ubouded sets ad theorems that are aalogous to P. P. Korovki theorem. Dokl. Akad. Nauk SSSR i Russia; Sov. Math. Dokl i Eglish 26. Doǧru O Gadjieva EA: Agirlikli uzaylarda Szász tipide operatörler dizisii sürekli foksiyolara yaklasimi II. Kizilirmak Uluslararasi Fe Bilimleri Kogresi Bildiri Kitabi Kirikkale 1998 Kousmaci: O. Dogru Türkçe olarak suulmus ve yayilamistir 27. Lopez-Moreo AJ: Weighted simultaeous approimatio with Baskakov type operators. Acta Math. Hug Leze B: O Lipschitz type maimal fuctios ad their smoothess spaces. Idag. Math Kolk E: Matri summability of statistically coverget sequeces. Aalysis Doǧru O Örkcü M: Statistical approimatio by a modificatio of q-meyer-köig ad Zeller operators. Appl. Math. Lett Duma O Orha C: Statistical approimatio by positive liear operators. Stud. Math

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