ON (p, q)-analog OF STANCU-BETA OPERATORS AND THEIR APPROXIMATION PROPERTIES

TWMS J. App. Eg. Math. V.8, N.1, 18, pp. 136-143 ON (p, q)-analog OF STANCU-BETA OPERATORS AND THEIR APPROXIMATION PROPERTIES M. MURSALEEN 1, TAQSEER KHAN, Abstract. I this paper we itroduce the (p, q) -aalogue of the Stacu-Beta operators ad call them as the (p, q)-stacu-beta operators. We study approximatio properties of these operators based o the Korovki s approximatio theorem ad also study some direct theorems. Also, we study the Voroovskaja type estimate for these operators. Keywords: (p, q)-calculus; Stacu-Beta operators; modulus of cotiuity; positive liear operators; Korovki type approximatio theorem; Lipschitz class of fuctios. AMS Subject Classificatio: 4A3, 41A1, 41A5, 41A36 1. Itroductio Durig the last two decades, there has bee itesive research o the approximatio of fuctios by positive liear operators itroduced by makig use of q-calculus. Lupas [1] was the first who used q-calculus to defie q-berstei polyomials ad later Phillips [] cosidered aother q-aalogue of the classical Berstei polyomials. Most recetly, Mursalee et al applied (p, q)-calculus i approximatio theory ad itroduced first (p, q)-aalogue of Berstei operators [13]. They ivestigated uiform covergece of the operators ad rate of covergece, obtaied Voroovskaya type theorem as well. Also, (p, q)-aalogue of Berstei-Stacu operators [14], Bleima- Butzer-Hah operators [16], Berstei-Schurer operarors [17] were itroduced ad their approximatio properties were ivestigated. Most recetly, the (p, q)-aalogue of some more operators have bee defied ad studied their approximatio properties i [1], [], [3], [5], [8], [11], [1], [15], [18], [19] ad [3]. So motivated by this, we itroduce the (p, q)-aalogue of the Stacu-Beta operators ad study their approximatio properties. We also study the Voroovskaja type estimate for these operators. We recall certai otatios ad defiitios of (p, q)-calculus. The (p, q)-iteger is a geeralizatio of q-iteger which is defied by [] p,q = p q, =, 1,,..., < q < p 1; N. p q The (p, q)-biomial expasio ad (p, q)-biomial coefficiets are defied by ( ) (ax by) p,q = q k(k 1) p ( k)( k 1) a k b k x k y k k p,q k= (x y) p,q = (x y)(px qy)(p x q y)...(p 1 x q 1 y). 1 Departmet of Mathematics, Aligarh Mus Uiversity, Aligarh, UP-, Idia. e-mail: mursaleem@gmail.com; ORCID: https://orcid.org/-3-418-47. Departmet of Mathematics, Jamia Millia Islamia, New Delhi-115, Idia. e-mail: tkha4@jmi.ac.i; ORCID: https://orcid.org/--714-7119. Mauscript received: August 19, 16; accepted: November 7, 16. TWMS Joural of Applied ad Egieerig Mathematics, Vol.8, No.1 c Işık Uiversity, Departmet of Mathematics, 18; all rights reserved. 136

M. MURSALEEN, TAQSEER KHAN: ON (P, Q)-ANALOG OF STANCU-BETA OPERATORS AND... 137 ( ) k p,q = k= [] p,q! [k] p,q![ k] p,q!. The defiite itegrals of a fuctio f are defied by a p k ( ) p k f(x)d p,q x = (q p)a q k1 f q k1 a, whe p q < 1 ad a f(x)d p,q x = (p q)a q k ( ) q k p k1 f p k1 a, whe q p < 1. k= There are two (p, q)-aalogues of the classical expoetial fuctio defied as follows p ( 1) e p,q (x) = [] p,q! x ad E p,q (x) = k= k= q ( 1) [] p,q! x. It is easily see that e p,q (x)e p,q ( x) = 1. For m, N, the (p, q)-beta ad the (p, q)-gamma fuctios are defied by ad Γ p,q () = B p,q (m, ) = x m 1 (1 x) m d p,qx p ( 1) E p,q ( qx)d p,q x, Γ p,q ( 1) = [] p,q! respectively. The two fuctios are coected through B p,q (m, ) = q m(m 1) p m(m 1) Γ p,q ()Γ p,q (m) Γ p,q (m ). (1) If p = 1, the the above otios of (p, q)-calculus (see [9], [1], [].) correspodig otatios of q-calculus.. Discussio Stacu itroduced i [4] the Beta operator as follows L (f, x) = 1 B(x, 1) t x f(t)dt. (1 t) x1 Aral ad Gupta [4] gave the q-aalogue of the Stacu-Beta operators as follows L (f, x) = K(A, [] q x) B([] q x, [] q 1) /A u []qx 1 (1 u) []qx[]q1 f(q[]qx u)d q u. reduce to the Here, we itroduce the (p, q)-aalogue of the Stacu-Beta operators ad study their approximatio properties. We also study the Voroovskaja type estimate for these operators. Let < q < p 1 ad x [, ). We itroduce the (p, q)-stacu-beta operators as follows 1 (f, x) = B p,q ([] p,q x, [] p,q 1) We have the followig auxilary result. u []p,qx 1 (1 u) []p,qx[]p,q1 f(p[]p,qx q []p,qx u)d p,q u. ()

138 TWMS J. APP. ENG. MATH. V.8, N.1, 18 Lemma.1. Let (f, x) be give by (). The for the polyomials t m, m =, 1,,... we have (t m, x) = p m(m 1) q m(m 1) ([] p,q x m 1)!([] p,q m)! (3) ([] p,q x 1)!([] p,q )! Proof. By (1), we have (t m, x) = B p,q([] p,q x m, [] p,q m 1) B p,q ([] p,q x, [] p,q 1) = p m(m 1) q m(m 1) ([] p,q x m 1)!([] p,q m)! ([] p,q x 1)!([] p,q )! which establishes (3). To examie the approximatio results for the operators i (), we eed the followig lemmas. Lemma.. Let (f, x) be give by (). The the followigs hold (i) (1, x) = 1, (ii) (t, x) = x, (iii) (t [], x) = p,q pq([] p,q 1) x 1 pq([] x. p,q 1) Proof. Puttig m = i (3), we have which proves (i). Puttig m = 1 i (3), Fially, puttig m = i (3), (1, x) = ([] p,qx 1)!([] p,q )! ([] p,q x 1)!([] p,q )! = 1, (t, x) = ([] p,qx)!([] p,q 1)! ([] p,q x 1)!([] p,q )! = [] p,qx([] p,q x 1)!([] p,q 1)! ([] p,q x 1)![] p,q ([] p,q 1)! = x. (t, x) = p 1 q 1 ([] p,q x 1)!([] p,q )! ([] p,q x 1)!([] p,q )! = 1 ([] p,q x 1)[] p,q x([] p,q x 1) pq ([] p,q x 1)[] p,q ([] p,q 1) = 1 ([] p,q x 1) pq ([] p,q 1) [] p,q 1 = pq([] p,q 1) x pq([] p,q 1) x, ad this proves (iii). This completes the proof of the lemma. Lemma.3. Let q (, 1) ad p (q, 1]. The for x [, ), we have (a) ((t x), x) =, (b) (t x), x) = ([]p,q pq[]p,qpq) pq([] p,q 1) x 1 pq([] x. p,q 1)

M. MURSALEEN, TAQSEER KHAN: ON (P, Q)-ANALOG OF STANCU-BETA OPERATORS AND... 139 Proof. Usig the Lemma (3), (i)-(ii), (a) is obvious. Also, which proves (b). (t x), x) = = t, x) x [] p,q x 1 pq([] p,q 1) x x t, x) x (1, x) = ([] p,q pq[] p,q pq) x 1 pq([] p,q 1) pq([] p,q 1) x, 3. Mai results This sectio is devoted to prove some direct theorems for the operators (f, x). By C B [, ), we deote the space of all real valued cotiuous bouded fuctios f o the iterval [, ) equipped with the orm f = sup f(x). x< If f C(I), δ > ad W = {h : h, h C(I)}, where I = [, ), the the Peetre s K-fuctioal is defied by K (f, δ) = if h W { f h δ h }. The there exists a costat C > such that (see [6]) K (f, δ) Cω (f, δ), (4) where ω (f, δ) is the secod order modulus of cotiuity of f C(I) defied by ω (f, δ) = sup <p<δ 1 sup f(x p) f(x p) f(x). x I The first order modulus of cotiuity of f C(I) is give by We prove the followig theorem. ω(f, δ) = sup sup f(x p) f(x). <p<δ x I Theorem 3.1. Let f C B [, ), x [, ) ad N. The there exists a costat C such that (f; x) f(x) Cω (f, δ (x)), where δ(x) = ([] p,q pq[] p,q pq) x 1 pq([] p,q 1) pq([] p,q 1) x. ad < p, q < 1. Proof. Let g W. By the Taylor s expasio we ca write g(t) = g(x) g (x)(t x) Operatig (., x) o both sides, (g; x) = g(x) t x (t u)g (u)du, t [, ). ( t ) (t u)g (u)du; x. x

14 TWMS J. APP. ENG. MATH. V.8, N.1, 18 So, By Lemma.3, we get (g; x) g(x) (g; x) g(x) By the defiitio of (f, x), The, Lp,q ( t (t u)g (u)du; x) x ( t ) t u g (u) du ; x x ((t x) ; x) g. ( ([]p,q pq[] p,q pq) x pq([] p,q 1) (f; x) f. ) 1 pq([] p,q 1) x g. (f; x) f(x) (f g; x) (f g)(x) (g; x) g(x) ( ) ([]p,q pq[] p,q pq) f g x 1 pq([] p,q 1) pq([] p,q 1) x g. Takig ifimum over g W, (f; x) f(x) K (f, δ(x)). Therefore (f; x) f(x) Cω (f, δ (x)). for every q (, 1) ad hece the proof is completed. Let B x [, ) deote the set of all fuctios f which are bouded by M f (1 x ), where M f is a costat depedig o f. By C x [, ), we deote the subspace of all cotiuous fuctios i the space B x [, ). Also we deote by Cx [, ), the subspace of all fuctios f(x) f C x [, ) for which x is fiite. 1x We prove the followig result. Theorem 3.. Let f C x [, ) be such that f, f C x [, ) ad p = (p ), q = (q ) with p, q (, 1) such that p 1, q 1 as. The uiformly o [, A], A >. [] p,q (L p,q (f; x) f(x)) = Proof. Usig the Taylor s formula, we ca get x(1 x) f (x) f(t) = f(x) f (x)(t x) 1 f r(t, x)(t x), (5) where r(t, x) is the remaider such that r(t, x) =. t x Operatig by L p,q (.; x) o both sides of (5), [] p,q (L p,q (f; x) f(x)) = [] p,q L p,q (t x; x)f (x) [] p,q L p,q ((t x) ; x) f (x) [] p,q L p,q (r(t, x)(t x) ; x).

M. MURSALEEN, TAQSEER KHAN: ON (P, Q)-ANALOG OF STANCU-BETA OPERATORS AND... 141 By the Cauchy-Schwarz iequality, we have L p,q (r(t, x)(t x) ; x) L p,q (r (t, x); x) L p,q ((t x) 4 ; x). (6) Notig that r (t, x) = ad r (., x) C x [, ), it follows from the Theorem 3.1 that Lp,q (r (t, x); x) = r (x, x) = (7) uiformly with respect to x [, A]. By (3), (6) ad (7) we immediately get [] p,q L p,q (r(t, x)(t x) ; x) =. Usig the Lemma.3, we get the followig ( [] p,q (L p,q (f; x) f(x)) = [] p,q f (x)l p,q ((t x); x) 1 ) f (x)l p,q ((t x) ; x) L p,q (r(t, x)(t x) ; x) This completes the proof of the theorem. = x(1 x) f (x). Next we preset the weighted approximatio theorem for operators (). Theorem 3.3. Let p = p, q = q be two sequeces such that < p, q < 1 ad p 1, q 1 ( ). The for f C x [, ). Proof. We show that Lp,q (f) f x =, Lp,q (t i ) x i x =, for i =, 1 ad. By usig (i) (ii) of Lemma., the coditios are easily fulfilled for i = ad 1. For i =, we ca write L p,q (t ) x x sup x [, ) sup x [, ) By the Korovki s theorem [7], we get [] p,q p q [] p,q p q p q ([] p,q 1) 1 x p q ([] p,q 1) 1 x. Lp,q (t, x) x x =. This completes the proof of the theorem. x 1 x 4. Coclusio Recetly, (p, q)-calculus has bee used i costructig (p, q)-aalogues of several classical operators ad ivestigated their approximatio properties. I this paper, we have itroduced the (p, q)-aalogue of the Stacu-Beta operators ad established some results o their approximatio properties by usig Korovki s approximatio theorem as well as direct theorems. We have also studied the Voroovskaja type estimate for our operators.

14 TWMS J. APP. ENG. MATH. V.8, N.1, 18 Competig iterests The authors declare that they have o competig iterests. Authors cotributios Both authors of the mauscript have read ad agreed to its cotet ad are accoutable for all aspects of the accuracy ad itegrity of the mauscript. Ackowledgemets The authors would like to exted their gratitude to the referees for their carefull readigs of the mauscript. Refereces [1] Acar T., (16), (p, q)-geeralizatio of Szász Mirakya operators, Math. Meth. Appl. Sci., 39 (1), pp. 685-695. [] Acar T., Aral A., Mohiuddie S. A., (16), O Katorovich modificatios of (p, q)-baskakov operators, J. Iequal. Appl., 16:98. [3] Acar T., Aral A., Mohiuddie S. A., Approximatio by bivariate (p, q)-berstei-katorovich operators, Ira. J. Sci. Techol. Tras. A Sci., DOI: 1.17/s4995-16-45-4. [4] Aral A., Gupta V., (1), O the q-aalogue of Stacu-Beta operators, Appl. Math. Letters, 5, pp. 67-71. [5] Cai Q. B., Zhou G., (16), O (p, q)-aalogue of Katorovich type Berstei Stacu Schurer operators, Appl. Math. Comput., 76, pp. 1-. [6] Devore R. A., Loretz G. G., (1993), Costructive Approximatio, Spriger, Berli. [7] Gadzhiev A. D., (1976), Theorems of the type of P.P. Korovki type theorems, Math. Zametki, (5), pp. 781-786. [8] Ilarsla H. G. I. ad Acar T., Approximatio by bivariate (p, q)-baskakov Katorovich operators, Georgia Math. J., DOI: 1.1515/gmj-16-57. [9] M.N. Houkoou, J. Dsir, B. Kyemba, (13), R(p, q)-calculus: differetiatio ad itegratio, SUT Jour. Math., 49(), pp. 145-167. [1] Lupaş A., (1987), A q-aalogue of the Berstei operator, Uiversity of Cluj-Napoca, Semiar o Numerical ad Statistical Calculus, 9, pp. 85-9. [11] Mishra V. N., Padey S., (16), O Chlodowsky variat of (p, q) Katorovich-Stacu-Schurer operators, It. J. Aal. Appl., 11(1), pp. 8-39. [1] Mursalee M., Alotaibi A., Asari K. J., (16), O a Katorovich variat of (p, q)-szász-mirakja operators, J. Fuct. Spaces, 16, Article ID 13553, 9 pages. [13] Mursalee M., Asari K. J., Kha A., (16), O (p, q)-aalogue of Berstei operators, Appl. Math. Comput., 66 (15), pp. 874-88 [Erratum: Appl. Math. Comput., 78, pp. 7-71]. [14] Mursalee M., Asari K. J., Kha A., (15), Some approximatio results by (p, q)-aalogue of Berstei-Stacu operators, Appl. Math. Comput., 64 (15), pp. 39-4 [Corrigedum: Appl. Math. Comput, 69, pp. 744-746]. [15] Mursalee M., Kha F., Kha A., (16), Approximatio by (p, q) -Loretz polyomials o a compact disk, Complex Aal. Oper. Theory, 1(8), pp. 175-174. [16] Mursalee M., Nasiruzzama Md., Kha A., Asari K. J., (16), Some approximatio results o Bleima-Butzer-Hah operators defied by (p, q) -itegers, Filomat, 3(3), pp. 639-648. [17] Mursalee M., Nasiuzzama Md., Nurgali A., (15), Some approximatio results o Berstei- Schurer operators defied by (p, q)-itegers, J. Ieq. Appl., 15:49. [18] Mursalee M. ad Nasiruzzama Md., (17), Some approximatio properties of bivariate Bleima- Butzer-Hah operators based o (p, q)-itegers, Boll. Uioe Mat. Ital., 1, pp. 71-89. [19] Mursalee M., Sarsebi A. M., Kha T., (16), O (p, q)-aalogue of two parametric Stacu-Beta operators, J. Ieq. Appl., 16:19. [] Phillips G. M., (1997), Berstei polyomials based o the q -itegers, The Heritage of P. L. Chebyshev, A. Numer. Math., 4, pp. 511-518. [1] Sadjag P. N., O the fudametal theorem of (p, q)-calculus ad some (p, q)-taylor formulas, arxiv: 139.3934 [math.qa]. [] Sahai V., Yadav S., (7), Represetatios of two parameter quatum algebras ad p, q-special fuctios, J. Math. Aal. Appl., 335, pp. 68-79.

M. MURSALEEN, TAQSEER KHAN: ON (P, Q)-ANALOG OF STANCU-BETA OPERATORS AND... 143 [3] Sharma H., O Durrmeyer-type geeralizatio of (p, q) -Berstei operators, Arab. J. Math., DOI 1.17/s465-16-15-. [4] Stacu D. D., (1995), O the beta approximatig operators of secod kid, Revue d Aalyse Numérique et de Thérie de l Approximatio, 4, pp. 31-39. M. Mursalee is a full Professor ad Chairma, Departmet of Mathematics, Aligarh Mus Uiversity (AMU), Idia. His research iterests are i the areas of pure ad applied mathematics icludig Approximatio Theory, Summability Theory, Operator Theory, Fixed Poit Theory, Differetial ad Itegral Equatios. He has published about 3 research papers i reputed iteratioal jourals ad eight books. He is member of several scietific committees, advisory boards as well as member of editorial board of a umber of scietific jourals. He has visited a umber of foreig uiversities ad istitutios as a visitig scietist/ visitig professor. He has a high umber of citatios ad presetly he has 148th Citatio Rakig (Mathematics) i the World by Web of Sciece, 17. He has bee awarded the Outstadig Researcher of the Year 14 of Aligarh Mus Uiversity. Recetly, he has bee recipiet of the Outstadig Faculty Research Award for the Year-18 by Careers36. Taqseer Kha is a Assistat Professor at the Departmet of Mathematics, Jamia Millia Islamia, New Delhi. He eared PhD i Mathematics joitly uder the guidace of Prof. M. Mursalee, AMU, Idia ad Prof. Ja Lag, Ohio State Uiversity (OSU), USA. He was a visitig US-Idia research STEM fellow at the OSU durig PhD, supported by the US-Idia Educatioal Foudatio (USIEF). He obtaied B.Sc (Hos) ad M.Sc. i Mathematics from AMU. He eared MS i Mathematics from the Cetral Europea Uiversity, Budapest, Hugary. He was the topper i B.Sc. (Hos.) ad topper (Gold Medalist) i M.Sc.. He has published eight (8) research papers i iteratioal jourals. His iterestig areas are Approximatio Theory, Operator Theory, Fuctioal Aalysis etc..